Combine the potatoes and brussel sprouts in a bowl. Set aside.
Melt 1 tablespoon butter in a medium-sized, heavy, non-stick skillet.
Saute the onion in butter until translucent.
Add 2 tablespoons of butter to pan, add potatoes and sprouts, mix thoroughly with onions, season with salt and pepper.
Press mixture down well to make a large pancake.
Let brown, undisturbed, for 6-10 minutes.
Once the bottom is browned, turn “cake” to brown other side.* When turning the cake, add another two tablespoons butter to the pan.
Once second side is browned, top with grated cheese, run under the broiler just until cheese is melted.
Serve in wedges straight from the pan.
[*Note: An easy way to turn the cake is to thoroughly loosen it around the edges and bottom with a spatula, set a plate over the skillet, hold the plate on the skillet and flip, turning the cake onto the plate. Return the pan to the heat, add the butter and slide the cake into the pan, browned side up.]
This recipe can also be made in individual patties. Simply form the patties with wet hands and brown as you would the cake.