Coat bottom and side of a 9-inch round springform pan with nonstick cooking spray. Line side with waxed paper, using spray to help adhere to pan. Trim paper to height of pan.

Finely crush 12 of the cookies in a food processor. Add milk; pulse just until mixture holds together. Set aside.

Unwrap 4 ice cream sandwiches. Working quickly, cut each in quarters. Stack strips of sandwiches on end, packing them snugly together, around waxed-paper-lined pan. Repeat with remaining sandwiches to form a stand-up edge. Spoon cookie crumbs into center of pan; press firmly over bottom. Freeze 1 hour.

Remove all 3 flavors of ice cream or frozen yogurt from freezer and let soften for 15 minutes at room temperature. Transfer mint ice cream to a small bowl and stir until good but firm spreading consistency. Repeat with vanilla and strawberry. Remove pan from freezer. Spread mint ice cream on bottom, then top with vanilla and strawberry, spreading all layers level. Top with thawed whipped topping, swirling top decoratively. Crush remaining 3 sandwich cookies and sprinkle over cake. Return to freezer; freeze overnight.

To serve, remove side of pan, then waxed paper. Cut into wedges.