Drain pinto beans and reserve liquid. Fork mash one can and leave one whole. Mix beans, bread crumbs, eggs, pepper and garlic in a medium bowl, adding enough bean liquid for mixture to hold together without being wet. Divide into 6 equal portions and flatten into 4-inch patties.
Mix mayonnaise and cocktail sauce and set aside.
Warm buns in a 300-degree oven, about 5 minutes. Meanwhile, heat oil in a large skillet over medium-high heat. Add patties and cook, turning only once, until a crisp brown crust forms on both sides, 6 to 8 minutes total. Top burgers with cheese slices, cover skillet, turn heat to low and let burgers continue to cook until cheese melts.
Spread cut side of each roll with the sauce. Set burger on bun bottom, top with sauerkraut, cap with bun top and serve.
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