How is Chauntecleer a round character in the Nun's Priest's Tale?

The Nun's Priest's Tale is a fable that allegorical. Though the characters are animals, the animals represent elements of human nature and the ultimate goal of the tale it to reveal a "universal truth" about human nature. 

Chauntecleer is best characterized by his physical appearance and abilities. You must also consider what characteristics are alluded to by the choice of animal, a rooster or cock, for the main character in the fable. When one thinks of a rooster, the descriptors often associated with this animal are egotistical, overly confident, and arrogance. In addition to the stereotypical character traits associated with a cock, we are told by the narrator that Chauntecleer is a prized cock because

"In all the realm of crowing without peer.

His voice was merrier than the play

Of the church's organ each holy day" (2850-2852).

He is also described as physically handsome

"His bill was black and like the jet it glowed,

His legs and toes like azure when he strode.

His nails were whiter than the lilies bloom," (2861-2863).

These traits ultimately cause Chauntecleer to ignore his beliefs that his own death is foretold in his dream. Although the narrator would contend that Chauntecleer's mistake was taking "his wife's advice, to his dismay"(3253), it is truly his belief that he is so wonderful and rare that he will be unharmed that leads to his fatal mistake. This is evident when Chauntecleer sings for the fox because of the fox's flattery. 

"This Chanticleer would not have tarried more
Once he espied the fox, had not the latter
Said, "Gentle sir, alas! what is the matter?
I am your friend--are you afraid of me?
I'd be worse than a fiend, most certainly,
To do you harm. And please don't think that I
Come here upon your privacy to spy;
The reason that I've come is not a thing
Except that I might listen to you sing. 
For truly you've a voice as merry, sire,
As any angel's up in heaven's choir.
Because of this, in music you've more feeling
Than had Boethius, or all who sing" (3282-3294).

As a result of Chauntecleer's ego, he is easily captured by the fox and taken off into the forest. Once there, we "For see how Fortune upsets suddenly /The hope and pride now of her enemy!" The fox's ego results in Chauntecleer's freedom because the fox cannot help but brag about his fortune. When the fox tries to entice Chauntecleer a second time, the cock will not be fooled. He tells the fox, "You shall no more, with words so flattering,/ Inveigle me to close my eyes and sing." He has learned his lesson. 

As a result, Chauntecleer may be considered a round character in the Nun's Priest's Tale because he learns from the errors of his ways and is not a victim of his ego a second time. A round character is one who develops throughout the course of a text; a static character remains unchanged. 

Why is Chanticleer a round character in "The Nun's Priest's Tale" from The Canterbury Tales?

Round characters, characters who are fully developed, have many sides to their personalities, so much so that they are often perceived by readers as realistic. The Nun's Priest's tale is much like a fable as the rooster, the hen, and the fox are anthropomorphized, or given human qualities, and the tale leads to a moral lesson.

Chanticleer, the rooster, is a vain male that has several hens, the favorite of which is "the winsome Pertelot." One day

This Chanticleer he groaned within his throat
Like man that in his dreams is troubled sore.

 Pertelot asks what is troubling him, and he replies,

Ah, madame,
I pray you that you take it not in grief,
By God. I dreamed I’d come to such mischief....

And, he tells his love about a beast that kills him. Pertelot scolds him for not being a man, though he has "a beard."  In reply to her, Chanticleer relates the story of a man who had a similar dream and was, in fact, killed.  But, foolishly he listens to Pertelot--"took the counsel of his wife, with sorrow"--and does not heed his dream.  Later, however, a fox slyly watches the chickens, and tells Chanticleer that he means no harm and only wishes to hear him sing. Thus deceived by this flattery, Chanticleer, like his father, closes his eyes to crow, and the fox grabs him by his throat and carries him off.  But, the quick-witted Chanticleer, realizing his foolishness, plays upon the pride of the fox and tells him to turn and taunt Mistress Pertelot and the others who pursue him by saying,

"‘Turn back again, presumptuous peasants all!
A very pestilence upon you fall!
Now that I’ve gained here to this dark wood’s side,
In spite of you this cock shall here abide.
I’ll eat him, by my faith, and that anon!’”

The gullible fox does just this, and when he opens his mouth, Chanticleer jumps out and flies upon a branch in a tree. Then the fox tries to trick the rooster, but the wiser Chanticleer tells him,

If you beguile me, having done so once,
You shall no more, with any flattery....

Clearly, then, Chanticleer is a round character; for, he has learned and developed from his earlier mistakes of letting flattery from Perteloe convince him to disregard his dream as well as the cajolery of the fox to get his to close his eyes as he sings.  Wiser now, Chanticleer will not be fooled a third time.

What details support Chauntecleer's round character in "The Nun's Priest's Tale"?

I would want to argue the central fact that supports the claim that Chanticleer (or Chauntecleer depending on the translation) is a round character is actually the way that he clearly shows he has learnt from his experiences with the fox by the end of the tale and also the way that he admits his own foolishness in giving in to the fox's flattery in the first place.

Let us just remind ourselves that a round character is a character who is shown to us in their full psychological complexity, rather than the rather two-dimensional nature of flat characters, who have but one or two character traits. We certainly see the full complexity of Chanticleer at the end of the tale when he talks to the fox when the fox asks him to come down:

You'll not, with your soft soap and flatteries,
Get me to sing again, and close my eyes!
To him who shuts his eyes when he should look,
And that on purpose, the Lord send bad luck!

Here we see Chanticleer being open about the way that he was almost fatally susceptible to "soft soap and flatteries," admitting his faults, but then also learning from this and integrating this into his character, showing that he is a round character rather than a flat one.