Preheat oven to 350° and line two 9-inch cake pans with parchment paper. Press 1 package cookie dough into bottom of each prepared pan. Bake until golden, about 20 minutes. Let cool in pans 10 minutes, then remove and freeze 30 minutes.
Remove cookies from freezer. Flip one over, flat-side up, and scoop on vanilla ice cream, smoothing top with a spatula to create an even layer. Sandwich with second frozen cookie, flat-side down, pressing to push ice cream out to the sides. Smooth ice cream along sides and press sides in chocolate chips until exposed ice cream is completely covered.
Set ice cream sandwich on serving plate and freeze until firm, 2 hours.
When ready to serve, remove from freezer to let soften, 10 minutes, then slice like a cake.