Rubuta ci gaba don wannan sakin layi - Brady ya ji daɗin aikin wasan hockey na matasa a cikin WHL tare da Medicine Hat Tigers. Bayan da ya zira kwallaye 81 a wasanni 72 kuma ya yi rajista fiye da mintuna 100 a shekarar 1985-86, New Jersey Devils sun zabi Brady a matsayin na uku a cikin 1986 NHL Entry Draft
Ka yi haƙuri. An dawo da shi zuwa ƙarami, sannan ya zama mai sana'a tare da AHL's Utica Devils a 1988/89. Ya fara wasan NHL a ranar 5 ga Oktoba, 1989.