Cream butter, shortening, sugars and vanilla with electric mixer fitted with paddle attachment until smooth and creamy. Gradually blend in eggs, beating well after each addition. Stir in combined flour, baking soda and salt until well blended. Stir in chocolate chips.
Portion dough onto parchment paper-lined sheet pans using #8 scoop, spacing about 2 inches apart. Combine chopped cookie pieces. Flatten dough slightly and evenly distribute cookie pieces over top of each cookie.
Bake in 375°F standard oven 16 to 18 min. or until golden brown on edges. Cool on pans 5 min. Remove to cooling rack. Cool completely before filling.
For each ice cream sandwich, place one #8 scoop ice cream on flat side of 1 cookie. Cover with second cookie, flat side down, to make sandwich. Freeze on sheet pans until ready to use. Let ice cream sandwiches stand at room temperature 5 min. before serving.