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1
In a 3-quart saucepan, melt the butter over medium heat. Add the onion and salt, toss to combine, and cover. Cook for 20 minutes, stirring several times to ensure that the onion does not stick to the pan. Add the beer, stir, cover, and cook for 10 minutes. If the onion is deeply caramelized and the liquid has mostly evaporated, it's perfection. If not, cook for a few minutes more. Set aside.
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2
In a large bowl, combine the lamb, garlic, smoked paprika, cayenne, and mint. Mix well. Set aside 4 tablespoons of the lamb mixture and divide the remaining lamb into four equal portions. Shape them into patties slightly larger than the width of the burger bun. Using your thumb, make a well in the center of each patty, going halfway through the lamb.
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3
Divide the goat cheese into four portions and place them in the thumb divots. Flatten each tablespoon of reserved burger mixture into a disc and place on top of the goat cheese. Pinch the edges to encase the cheese in the lamb. Season both sides of each patty with salt and pepper.
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4
In a large skillet, melt the butter with the olive oil over medium heat. Fry the burgers for 4 to 5 minutes per side for medium doneness.
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5
Place the burgers on the buns and top each with 1 tablespoon mayo, some caramelized onions, bacon, and 1/4 cup arugula.
