Cika wannan jumla: An tsara shi a 68th a cikin
1988 NHL Entry Draft da New York Rangers, Amonte ya fi sani da lokacinsa a matsayin tauraruwa mai zira kwallaye tare da Chicago Blackhawks da kuma wakiltar Amurka a wasan kasa da kasa. Amonte ya fara bugawa a wasan karshe na 1991. Ya burge a matsayin rookie, ya zira kwallaye sama da 30 kuma ya zama na uku a zaben Calder Memorial Trophy. Ya buga wasanni uku tare da Rangers, inda ya zira kwallaye 84, kafin a siyar da shi zuwa Chicago Blackhawks tare da wasanni bakwai da za a yi a kakar wasa ta 1993-94, shekarar da Rangers ta ci gaba da lashe Stanley Cup. Ya sami daraja a Chicago inda aka fara haɗuwa da tsohon abokin aikin Thayer Academy Jeremy Roenick, inda ya zira kwallaye akalla sau 30 sau shida kuma akalla sau 40 sau uku, gami da samun tsawon kakar wasa biyar wanda bai rasa wasa daya ba.