Special equipment: four 6-inch cake pans; a 5-inch ring cutter or a paring knife; a piping bag fitted with a round tip; an offset spatula; two to four 4-inch wooden skewers
For the cake: Preheat the oven to 350 degrees F. Spray four 6-inch cake pans with cooking spray.
Prepare the cake mix according to the package directions. Divide the batter among the prepared pans and bake until a tester inserted in the center comes out clean, about 35 minutes. Cool on a rack 10 minutes, then unmold and cool completely.
For the buttercream frosting: Beat the sugar, butter and salt with an electric mixer on medium speed in a large bowl until incorporated. Add the vanilla, increase the speed to medium high and mix until smooth. Mix in enough milk to make it spreadable. Reserve 1/2 cup frosting in a piping bag fitted with a round tip. Mix food coloring into the remaining frosting to desired color.
For the decorations: Cut off the domed tops of the cake rounds with a serrated knife so they are completely flat (save the trimmings for another use). Using a paring knife or 5-inch round cutter, cut a 5-inch circle from the center of one cake round, making sure to keep the outer ring intact (this will be the rim of the mug). Trim the cutout circle to a thickness of 1 inch, then use a paring knife to cut out a handle that is about 4 inches long and 2 inches wide. While you assemble the cake, freeze the handle on a small plate until very firm (this will make it easier to frost).
Place a cake round cut-side up on a cake stand or serving platter. Spread with a thin layer of frosting almost to the edge. Top with another cake round and spread with a thin layer of frosting. Repeat with a third round. Top with the cake ring. Frost the entire cake with a thin layer of frosting to make a crumb coat (you don’t need to frost the inside of the ring). Spread a thin layer of frosting on the handle. Freeze the cake and handle until the frosting is set and the handle is firm, about 30 minutes.
Frost the cake–excluding the center cutout on the top–with more frosting. Smooth the sides and rim with an offset spatula. Frost the handle.
Hold the handle gently against the cake and insert two to four 4-inch skewers to attach it. Smooth over any smudges in the frosting on the handle. Pipe a stripe of the reserved white frosting around the top and bottom of the mug (or wrap with strips of sour candy rope).
When ready to serve, fill the cavity at the top of the cake with the pudding and spread flat to the rim with an offset spatula. Just before serving, top with whipped cream, leaving a rim of pudding showing.