In a saucepan, melt the butter and olive oil over medium heat. Allow the butter to brown, then reduce the heat to low and add the chopped onions. Cover and cook over very low heat, stirring occasionally, until soft and caramelized, about 20 minutes.
Meanwhile, shape the seasoned ground beef into 2 patties and set aside.
In a skillet, melt the ghee over medium-high heat. Cook the hamburger patties for about 3 minutes per side, adding the cheese to each patty once flipped. Allow the cheese to melt.
Preheat the oven's broiler and toast the buns until crisp, 1 to 2 minutes.
Assemble the burgers by placing the patties on the bottom buns, topping with caramelized onions, and finishing with a spread of butter on the freshly toasted top bun.
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