Preheat the oven to 350 degrees F (175 degrees C). Combine Cheddar and mozzarella cheeses in a bowl.
Heat a large skillet over medium-high heat. Cook and stir sausage in the hot skillet until browned and crumbly, 5 to 7 minutes. Set sausage aside.
While sausage is frying, bring a pot of water to a boil. Cut the tops off of the green peppers and trim the insides. Parboil the peppers in the pot, 4 to 5 minutes. Remove to cold water, dry, and lightly salt the insides.
Heat oil in the same skillet over medium-high heat. Fry the potatoes and onion in the hot oil, breaking up and turning periodically, until potatoes are golden brown and crispy, 5 to 7 minutes. Add sausage, 1/2 of the cheese blend, eggs, salt, and pepper. Turn off heat, mix gently but thoroughly, and let sit for 5 minutes. The eggs will not be fully cooked at this point, but will finish while baking.
Give the potatoes, eggs, and sausage blend one final mix and then gently stuff into the green peppers about 90% full, leaving space for reserved cheese blend. Do not pack tightly. Place in a baking dish and cover.
Bake in the preheated oven for 15 minutes. Uncover and continue to cook until eggs are set and peppers are tender, about 10 minutes. Top with reserved cheese blend and turn on the oven’s broiler. Broil until cheese is browned, 4 to 5 minutes.