First of all, you need to be sure you copied this sentence correctly because it contains a grammatical error. If there is more than one contestant, then there is more than one winning essay, right?
Now, as to the clause, what makes a clause subordinate or independent? The test is simple. Would this clause make a proper sentence if you put it on its own? Of course not! "Whose essay won" cannot stand alone. Now, there is a catch to this. If you make this clause a question, it can become an independent clause because it can stand alone as a question. It has a subject "whose essay," a verb, and it expresses a complete idea. So, whether this clause is subordinate or independent is a function of whether or not you are permitted to punctuate it appropriately.