For the burger: Preheat the oven to 350 degrees F.
Place the bacon on a rimmed baking sheet and bake until golden brown but not completely crisp, 15 to 20 minutes. Keep warm for the burgers.
For the beer cheese sauce: Meanwhile, melt the butter in a heavy-bottom saucepan over medium heat. Add the onions and cook until soft, 2 to 3 minutes. Add the flour and cook, stirring, until the roux starts to brown, about 2 minutes. Whisk in the beer until smooth.
Reduce the heat and cook until the beer begins to reduce and the alcohol is cooked out, 4 to 5 minutes. Add the cream and garlic and continue to cook, stirring continuously, until the sauce thickens, about 10 minutes. Remove from the heat and season with 1/2 teaspoon salt and some pepper; gradually whisk in the cheeses until melted completely. Keep warm. Note: This recipe makes about 4 cups of cheese sauce, more than needed for the burgers. Refrigerate the extra for up to 3 days.
To make the burgers, cut the butter into four 1/2-inch thick pieces and set aside, taking care to keep them cold. In a medium bowl, gently fold the ground beef with 1 to 2 teaspoons salt and some pepper to season, and divide into four equal portions. Form each portion of meat into a ball and press your thumb into the center. Place a cold slice of butter into each indent and cover over with some meat, pressing each ball into a patty so the butter is in the center. Hold the patties on a plate for cooking.
Heat a grill pan over medium high heat, and brush with oil. When the pan is hot, add the patties, spacing them evenly. Cook on one side until browned, about 5 minutes, and then flip the patties and cook to desired doneness, adding the cheese slices in the last 2 minutes.
Butter both sides of the pretzel buns and place on the grill pan, cut-side down. Grill until golden brown and toasted. Place the buns onto plates, and place the patties onto the toasted buns. Top each burger with 2 bacon slices and 1/4 cup beer cheese sauce. Serve immediately.
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