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.