Position an oven rack in the center of the oven and preheat to 375 degrees F.
Remove 1/4 inch of the crust from one end of the bread and enough of the crust from the other end so you have a 12-inch loaf. Reserve the ends for another use. 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 1/3 cup of the potato chips and 1 slice of American 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, stir together the mayonnaise, mustard, celery, shallot, some hot sauce and a large pinch of salt in a medium bowl until combined. Fold in the tuna. Season with additional salt and hot sauce if needed.
Fill each bread box in this order: 1/2 cup of the tuna salad, 4 pickle slices, another 1/2 cup of the tuna salad, 1 slice of tomato and 2 additional slices of American cheese. Return the boxes to the oven and bake until the cheese is melted, about 5 minutes. Top with the sprouts, if using, cut in half or leave whole and serve immediately.
Don't worry if you accidentally make a hole at the bottom of the bread when carving out the boxes--just fill the hole with a loose piece of bread.
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