Preheat the broiler. Melt 2 tablespoons butter in a large cast-iron skillet over medium heat. Add the onions and cook until translucent, 6 to 8 minutes. Add 1/3 cup water, cover and continue cooking until the onions are golden brown, about 15 more minutes. Season with salt and pepper. Transfer the onions to a bowl and wipe out the skillet.
Divide the meat into 6 loose balls, then gently flatten into thin patties. Heat the skillet over high heat. Add the patties to the hot skillet, smash them flat with a spatula and season with salt and pepper. Cook, flipping once, until well done, about 3 minutes per side.
Meanwhile, toast the buns under the broiler. Put the patties on the bottom buns and top each with 1/4 cup sauteed onions. Smear 2 tablespoons butter on the cut side of each top bun, then put on top of the burgers. Slice in half and serve immediately.
Photograph by Tina Rupp
Recipe adapted from Solly's Grille, Milwaukee for Food Network Magazine