Top one slice of bread with a quarter of the shredded cheese, half of the caramelized onions, a couple cranks of coarse black pepper, and another quarter of the shredded cheese. Place another slice of bread on top of the cheese, spread with about a quarter of the butter and lightly sprinkle with a little kosher salt.
Repeat with the remaining bread, cheese, onions, and quarter of butter.
Bring a large skillet to medium heat and add the sandwiches butter side down. Spread the remaining butter onto the slices of bread that are facing up and lightly sprinkle with a little kosher salt.
Cook until the bread is beautifully browned and the cheese is melted, about 3 minutes per side.
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