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Divide ground bison into 4 equal portions and place large indentations in the center of the patties, forming small “bowls.” Sprinkle ½ -1 tablespoon of blue cheese into the center of each patty. Fold the outer edges over the center, encasing the cheese. Set aside.
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Preheat grill or grill pan to medium-high heat, open buns and place on the grill about 2 to 3 minutes to toast. Set aside.
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Oil grill and add the onion slices. Grill on both sides until grill marks form and the onion is slightly softened, about 8 minutes total. Transfer to a plate and set aside.
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Season patties with salt and pepper. Place them on the grill and cook about 3 minutes per side for medium or about 2 minutes per side for medium rare.
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Spread each bottom hamburger bun with mayo and mustard and top with the arugula, onion, burger, bacon, tomato and the top bun. Serve immediately.