The 20-year-old had made just one substitute first-team appearance before this campaign, but he has quickly established himself as a key player at San Siro, scoring 15 goals in all competitions.
"[Milan coach Gennaro] Gattuso has given me a little bit of his character, but I always had a competitive nastiness and a desire to score," the forward told a news conference.
"I didn't expect a season like this. Last year I was in the [Milan] Primavera and I honestly didn't think I'd get to this point, [but] I showed what I could do and trained hard. I would like to thank all the the coaches that helped me to be here, from those in the youth team to [Vincenzo] Montella and Gattuso. Thanks to you all.
"Dreams never end, especially for us youngsters. I hope to make my debut soon.
"I still haven't achieved anything, though. My parents help me a lot and keep my feet on the ground. This is an opportunity and I'll try not to waste it. I want to train well and improve. This is a starting point and I want to do more and more."
Italy face Argentina on Friday and Cutrone is relishing the prospect of taking on Barcelona great Lionel Messi.
"It will be an honour to face him," Cutrone added. "He is football."