Cook the bacon in a skillet over medium heat until crisp, 5 to 6 minutes. Drain, cool slightly, and break each slice into two shorter pieces.

Whisk the eggs with salt and pepper. Melt a teaspoon of butter in a nonstick skillet over medium-low heat, pour in the eggs, and cook gently, stirring just until set but still glossy, 2 to 3 minutes. Shape into a rectangle roughly the size of the rolls.

Preheat the oven to 375°F. Slice the rolls in half horizontally, keeping them connected. Place the bottom slab in a baking dish, brush with half the melted butter, and layer the eggs, cheese slices, and bacon. Top with the roll tops, press down gently, brush with remaining butter, and sprinkle with seasoning.

Cover loosely with foil and bake 6 to 7 minutes. Uncover and bake 5 to 6 minutes more until tops are golden and cheese is melted.

Let rest 2 minutes, then slice between the rolls along the seams. Serve warm.