Light a grill. Season patties generously with salt and pepper. Grill the patties over moderately high heat until lightly charred on the outside and medium-rare within, about 5 minutes per side. Just before the burgers are done, lay a slice of cheese on each one to melt.
Spread the cut sides of the buns with the melted butter and toast on the grill. Transfer the burgers to the buns. Top with the Griddled Onions, sauerkraut and Red Chili and serve.