Butter and flour an 8-inch-round cake pan and a 1.5-quart ovenproof bowl. Make the cake batter; divide between the pan and the bowl. Bake at 350 degrees F until a toothpick comes out clean, about 35 minutes for the pan, 40 to 45 minutes for the bowl. Let cool, then unmold the cakes onto a rack.
Microwave the white chocolate at 50% power in 15-second intervals, stirring, until melted, about 2 minutes. Pour onto a parchment paper-lined baking sheet and spread to 1/4 inch thick with an offset spatula. Sprinkle with the sanding sugar. Chill until set, about 10 minutes.
Remove the chocolate from the refrigerator. Break most of it into rough 1/2-to-1-inch pieces to form the ice that covers the igloo.
Use a paring knife to cut a few even 1/2-inch tiles from the remaining chocolate (these are for the tunnel entrance).
Assemble the igloo: Mix 2 1/2 cups frosting with the blue food coloring. Spread some blue frosting on the flat cake, then cover with the bowl-shaped cake.
Spread some frosting on the cake where you want the tunnel to go, then attach a doughnut half, cut-side down. Attach the other half with frosting to extend the tunnel.
Cover the entire cake with a thin layer of blue frosting (this is called a "crumb coat," which is like a primer; it doesn't have to be perfect). Chill, uncovered, until the frosting sets, about 15 minutes.
Spread the remaining blue frosting evenly over the cake and doughnuts. Arrange the white-chocolate pieces on the igloo in a mosaic pattern. Place the even white- chocolate tiles around the tunnel entrance.
For the trees, mix 3/4 cup vanilla frosting with the green food coloring. Spread on the cones, flicking the frosting with the tip of an offset spatula to form branches. Arrange the trees around the igloo and dust with confectioners' sugar.