Position an oven rack in the center of the oven and preheat to 375 degrees F.
Remove 1/4 inch of the crust from the ends of the loaf, then cut the loaf crosswise into 4 even pieces. Use a fork to score one cut end and gently dig a square hole into each piece of bread to make a box, leaving a 1/2-inch layer of bread around the walls and bottom (see Cook's Note). Save the removed crumbs for another use.
Microwave the butter and a large pinch of salt in a small microwave-safe bowl until melted, about 1 minute. Brush the bread boxes inside and out with the melted butter. Put 1 slice of cheese at the bottom of each bread box. Put the boxes open-side up on a baking sheet and bake until the cheese melts and the inside of the bread is lightly browned and crispy in parts, about 10 minutes.
Meanwhile, heat a large non-stick skillet over medium heat and add the bacon. Cook, stirring occasionally, until the bacon is crispy, 12 to 15 minutes. Use a slotted spoon to transfer the bacon onto a paper-towel lined plate. Reserve the bacon fat in the skillet and adjust the heat to low. Whisk the milk and eggs with a large pinch of salt and pepper and add to the skillet. Continue stirring the eggs, scraping the sides of the skillet as needed, until small curds form and the eggs are just set, 6 to 8 minutes.
Divide the hot eggs between the 4 boxes and top with the remaining cheese. Return to the oven and bake until the cheese is melts, about 5 minutes. Top with the crispy bacon and serve hot with hot sauce and additional salt and pepper.
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