Put the potatoes in a saucepan and add enough water to cover generously. Salt the water, bring it to a simmer over medium heat, cover partially, and cook until the potatoes are tender enough to stick a fork into but still on the firm side of done, about 20 minutes. Drain the potatoes and let cool. (You can boil the potatoes up to a day ahead of time and keep them in the refrigerator if you like.)
While the potatoes are cooking, clarify the butter: Heat the butter in a small skillet over medium heat until it foams and starts to turn dark blond on top, 3 to 5 minutes. Remove from the heat and let the butter sit for a few minutes. Then tilt the skillet toward you and gently spoon off and discard the layer of foam. Pour the clear golden butter into a bowl, discarding the dark dregs at the bottom of the skillet.
Coarsely grate the potatoes, skins and all, onto a baking sheet, making sure to keep them loose and not to pack them down. Combine 1/2 teaspoon salt and the black pepper in a small bowl.
Heat a 10-inch cast-iron skillet over medium-high heat, and add half of the clarified butter. Add a third of the potatoes in an even layer, keeping them loose. Top with half of the onions and season with some of the salt and pepper mixture. Add another third of the potatoes and the rest of the onions, and season again with salt and pepper. Add the remaining potatoes, season again, and then reduce the heat to medium and cook the potato cake until the bottom turns dark amber brown, about 15 minutes. (Peek on the sides: If it's browning too quickly, turn the heat down to medium-low.)
Put a large plate upside down over the skillet, and using two thick oven mitts, grab the sides of the skillet and turn it upside down, releasing the hash-brown cake onto the plate. Put the empty skillet back on the burner and add all but 1 tablespoon of the remaining clarified butter. Slide the potato cake back into the skillet. Cook until the underside turns dark amber brown, 10 to 15 minutes.
While the potato cake is cooking, fry the peppers: Heat a large saute pan over medium heat and add the remaining tablespoon of the clarified butter. Add the peppers and 1/4 teaspoon salt, and cook, flipping them often, until they blister and feel tender throughout, about 10 minutes.
To serve, lay the large plate upside down over the skillet, and using oven mitts, invert as before. (If the first side was prettier, invert the cake again onto another serving plate.) Top with a jumble of fried peppers. Cut into wedges and serve immediately.
Adapted from "The New Midwestern Table" by Amy Thielen (c) Clarkson Potter, 2013. Provided courtesy of Amy Thielen. All rights reserved.