Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain. Slice potatoes when cool enough to handle.

Melt 2 tablespoons butter in a large skillet over medium heat; cook and stir onion in hot butter until slightly browned, about 10 minutes. Melt remaining 2 tablespoons butter in skillet; add sliced potatoes. Continue cooking until potatoes begin to brown, about 10 more minutes; stir often.

Beat eggs in a bowl; pour beaten eggs over potatoes and let cook until egg begins to firm on the bottom, about 5 minutes. Carefully work a spatula underneath the eggs and potatoes and flip. Cook until eggs are firm and set throughout, 3 to 5 more minutes. Season with salt and black pepper.

Lay American cheese slices over egg-potato mixture and let melt before serving.