Slice the tops off the cakes to smooth the surface.
Stack the cakes and spread some buttercream between each layer. Smooth the sides of the cake so that the buttercream filling is flush with the sides. Chill the cake in the fridge for 30 minutes.
Using a serrated knife, carve the base of the cake so that it curves inwards, like a mug. Stick 4 skewers vertically into the cake. This will help to stabilize it.
Coat the cake in a thin layer of buttercream, called a crumb coat. This will catch any excess cake crumbs. Chill the cake in the fridge for 1 hour.
In the meantime, shape the fondant into a mug handle. Once you are happy with the shape, insert a skewer into each end of the handle. Place the handle on a plate lined with plastic wrap and allow it to fully harden, 1-2 hours.
Coat the cake in a thick, generous layer of buttercream. Place the remaining buttercream into a piping bag (or Ziploc bag) and snip the end to create a medium-sized hole. Pipe 3-4 loops of buttercream around the rim of the mug, stacked into a pyramid. Use your cake knife or spatula to smooth the edges until it looks like the rim of a mug! Place the cake in the fridge while you make the ganache.
To make the ganache, combine the white chocolate and 1/3 cup whipping cream in a bowl. Whisk until fully combined – this may take a minute while the chocolate adjusts to the heat and consistency of the cream. Repeat with the milk chocolate and 1/3 cup whipping cream. Allow the 2 ganaches to cool at room temperature until they are no longer warm, but still liquid enough to pour.
Stick the mug handle into the side of the cake. Be careful not to tip the cake over!
Pour the milk chocolate ganache into the chilled mug cake and smooth the surface. Drizzle some white chocolate ganache on top and swirl with a butter knife, until your desired pattern is achieved. Chill the cake for 30 more minutes, allowing the ganache to set. Then slice and enjoy!