Add the 4 tablespoons butter to a large heavy-bottomed saucepot over medium-high heat. In batches, fry the ham until crisp, 8 to 10 minutes. Drain on paper towels.
Reduce the heat to medium-low and add half of the fried ham back to pot; reserve the remainder for garnish. Whisk in the flour and cook for 4 to 5 minutes to create a medium-dark roux.
Slowly whisk in the chicken stock, bring to a low simmer and cook until the gravy has thickened, 5 minutes. Season with the paprika and 1 teaspoon salt and pepper.
Heat the remaining 2 tablespoons butter in a large skillet over medium heat. Add the pearl onions, and sear briefly. Add 1/2 teaspoon salt; cook on all sides for 2 minutes. Add 1/2 cup water, reduce the heat to low and simmer until the liquid has mostly evaporated and onions are tender, 4 minutes.
For each serving, place one-quarter of the tater tots on a heat-proof serving plate. Scatter 1/4 cup of the mozzarella pieces over the tater tots. Add 1 cup of the ham gravy, one-quarter of the braised onions and 1/2 cup of the pepper jack cheese. Place under the broiler; broil until the cheese is melted, about 3 minutes. Sprinkle with some of the sliced banana peppers and reserved fried ham.
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