Cook the onions, potatoes and garlic: In a large bowl, toss the onions in 2 tablespoons oil, season generously with salt and pepper and spread on one side of a sheet pan. In the same bowl, toss the potatoes in 1 tablespoon oil, season generously with salt and pepper and spread on the other side of the sheet pan. Toss the garlic in the same bowl with a little oil, season with salt and pepper and add to the sheet pan. Bake in the center of the oven until completely tender when pierced with the tip of a knife, about 40 minutes. Set aside to let cool.
Cook the peppers: In a large cast-iron skillet over medium heat, heat 3 tablespoons oil until hot. Add the peppers with a generous pinch of salt and cook over medium heat until tender and slightly browned, 5 to 8 minutes.
Cook the home fries: Add the onions and garlic to the peppers and stir to combine. "Crush" and break the potatoes into pieces with your hands or the bottom of a pan and add them and any juices from the pan to the peppers and onions. Season with salt and red pepper flakes. Cook the potato mixture over medium-high heat until the bottom is browned, 2 to 3 minutes, then flip and cook for another 2 to 3 minutes.
Stir in the Parmesan and vinegar. Top with the Cheddar. Place the skillet in the oven and bake until the cheese is browned, 8 to 10 minutes. Season with salt just before serving.
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