Place chopped almonds on a tray and toast in preheated oven until lightly colored and fragrant, 5 to 8 minutes. Watch closely to prevent burning; if there is a strong odor of nuts in your kitchen, they'll be burnt by the time they cool. Let cool completely.
Crumble the amaretti into 1/2 inch pieces by wrapping in a tea towel and tapping lightly with a rolling pin or mallet. Do not crush into small pieces. Place crumbled cookies in a medium bowl and sprinkle with rum. Toss with almonds and chocolate chips. Set aside.
In a large bowl, beat cream with an electric mixer until thickened. Pour in brandy and beat until soft peaks form. Spoon whipped cream over cookie mixture. In same bowl, beat ice cream just until softened (with a flat whip, if available). Fold cookie and cream mixture into ice cream. Freeze until ready to serve.