For filling, in a small bowl stir together cream cheese and sugar. In a medium bowl use a wooden spoon to stir ice cream just until it begins to soften; fold cream cheese mixture into ice cream. Place in freezer while preparing the cake.
Use a serrated knife to cut off the top 1/2 inch of the cake; set aside. Hold the knife parallel to the center hole of the cake and cut around the hole, leaving about 3/4-inch thickness of cake around the hole. Cut around the outer edge of the cake, leaving an outer cake wall about 3/4 inch thick. Use a spoon to remove center of cake, leaving about a 3/4-inch-thick base. (Reserve scooped-out cake for another use.)
Spoon filling into hollowed cake. Arrange sliced strawberries on the filling. Replace the top of the cake. Cover and freeze for at least 6 hours or up to 24 hours.
To serve, in a small saucepan heat ice cream topping until drizzling consistency; drizzle over cake. Slice cake with a serrated knife.