Butter and flour a 10-cup bundt pan or square baking dish. Slice 1-inch off tops of rolls. Tear tops into 1-inch pieces; set aside. Place bottoms of rolls, cut side up, in prepared pan. Combine coffee cake batter mix (reserve cinnamon swirl packet), water, instant espresso, oil, and 2 eggs in a large bowl. Beat until well blended. Pour half of batter over rolls. Sprinkle with contents of cinnamon swirl packet.
Whisk remaining 1 egg, milk, and sugar in another bowl to blend. Add reserved roll tops to milk mixture; set aside until milk mixture is absorbed. Pour bread mixture over cake batter in pan, and then pour remaining batter over. Using a skewer or wooden chopstick, swirl batters together. Bake for 45 minutes, or until a toothpick inserted near the center of cake comes out clean. Cool cake in pan on a cooling rack for 25 minutes. Invert cake onto cooling rack; remove pan. Place cake right side up on a platter and serve warm. For a cute presentation, serve slices in coffee mugs.