You can use that phrase with no article, with the indefinite article or with the definite article.
If you are in a high school around exam time, and someone asks "why don't you go sit in the lunch room", you could respond with "the last thing I want to do is listen to adolescent agonizing". In this case, "agonizing" is used like a noun, and adolescent is used to modify it.
Then, if someone says "No, you need go talk to Tommy about his exam anxiety". You could response "I have better things to do than to listen to an adolescent agonizing about exams". Here, "adolescent" is a noun, and the "agonizing" is a verb.
If you avoid talking to Tommy, but another colleague comes by, looks over at Tommy, pointing him out to you and says "listen to the adolescent agonizing - it must be exam time". Now, depending on how you emphasize things, there are two choices. In one case, your colleague is truly talking about Tommy, in which case "adolescent" is the noun, and the "agonizing" is a verb. But, he/she could be speaking in a more general sense, and it's closer to the non-article version, with "agonizing" as a noun and "adolescent" modifying it.