Let all ice cream stand at room temperature for about 15 minutes or until soft enough to spread. Meanwhile, line a 3-quart metal bowl with plastic wrap, allowing extra wrap to hang down outside of bowl.
Spoon strawberry ice cream into bottom of bowl. Arrange brownies on top in a single layer; press on them to compress with ice cream in an even layer. Spoon in pistachio ice cream, smoothing top evenly with a spoon. Sprinkle with chopped cookies. Spoon coffee ice cream on top of cookies. Wrap overhanging plastic over ice cream and compress again. Freeze until firm, at least 3 hours.
To unmold, let bowl stand at room temperature for 15 minutes. Wrap a towel soaked in hot water around bowl. Unwrap top of bombe, invert bowl onto a round platter and pull on plastic until bombe is released. (If necessary, resoak towel in hot water.) Remove and discard plastic wrap and return bombe to freezer.
In a large bowl, whip egg whites with an electric mixer at medium speed until soft peaks form. Gradually sprinkle in sugar while mixing at low speed. Once all sugar has been added, beat at high speed until stiff. Whip cream in a separate bowl until stiff but not dry. Fold cream into egg white mixture. Spread cream mixture on top of bombe. Garnish with chocolate sprinkles. To serve, cut with a large, sharp knife.