Preheat the oven to 350 degrees and grease an 8 inch round cake pan. Line the bottom with a parchment round if desired.
Cream together the butter, oil and sugar until light and fluffy. Add the egg, blending well.
Beat cream cheese and butter until fluffy, about 2 minutes on med/high speed with an electric mixer.
Crumble cooled cake by hand or in a food processor until fine crumbs. Pour the crumbs into a mixing bowl and add 1/2 - 3/4 cup frosting (the rest can be refrigerated and stored for another use).
Mix with the paddle attachment of a mixer on low speed until mixture holds together and has the consistency of play-doh or wet sand. Alternatively mix well by hand with a rubber spatula or wooden spoon.
Spread cake/frosting mixture into an 8 inch round cake pan and smooth, pressing down as you smooth. It should be about 1 inch thick.
Line a baking sheet with wax paper or parchment and using the round cutter, cut out about 28 rounds, reforming scraps as needed. Lay the rounds on the baking sheet and place in the freezer for 15 minutes or in the refrigerator for 1-2 hours.
Melt candy wafers according to package instructions. Dip the end of each lollipop stick into the coating, then stick in each round and allow to harden before dipping.
Sprinkle with edible confetti or sprinkles and place a candle in the middle (if using) and stick the pop in the styrofoam block to dry.
Melt colored candy wafers according to package instructions. Use the melted wafers to pipe an upper and lower decorative border on each pop to resemble a birthday cake. Use a disposable pastry bag with just the end snipped or fit it with a #2 round tip.
Pops can be stored at room temperature for 3 days.
*1 1/2 inch round cutter, styrofoam block, lollipop sticks, disposable pastry bag (#2 pastry tip - optional), trimmed birthday candles (optional)