Peppermint Ice Cream Candyland Cake

Preheat oven to 350 °F. Butter 15x10x1-inch baking sheet; line with parchment. Butter parchment. Whisk first 4 ingredients in large bowl. Melt butter in medium saucepan. Remove from heat. Add chocolate chips; stir until melted. Whisk in 3/4 cup water, cocoa, and vanilla until blended. Whisk in egg. Add to dry ingredients; whisk to blend. Pour batter onto prepared baking sheet. Bake cake until tester inserted into center comes out with some moist crumbs attached, about 22 minutes. Cool cake on baking sheet on rack 15 minutes. Run knife around pan sides to loosen cake. Turn cake out onto sheet of foil; cool. Cut cake crosswise into 3 equal strips. Freeze cake strips 1 hour. Place 1 cake strip on platter. Working quickly, spoon 3 1/2 cups peppermint ice cream in dollops over cake; spread evenly to edges. Top with second cake strip; spoon 3 1/2 cups ice cream in dollops over cake; spread evenly to edges. Top with third cake strip; freeze 1 hour. Spread enough remaining ice cream over top and sides of cake to cover generously (about 6 cups). Freeze until firm, about 3 hours.