How to Make It
In a large bowl, lightly mix the lamb with the herbes de Provence and paprika and season with salt and pepper. Loosely form the mixture into four 5-inch patties and make a slight depression in the center of each one to help the patties retain their shape while they cook.
In a large cast-iron skillet, heat the olive oil. Cook the patties over moderately high heat until nicely charred outside and medium-rare within, 5 minutes per side. Transfer the burgers to a plate. Pour off all but 2 tablespoons of the fat and return the skillet to moderate heat. Toast the buns, cut side down, until golden, about 2 minutes. Transfer the buns to a plate.
In the same skillet, melt the butter. Add the shallots and cook over moderately high heat, stirring, until tender, about 2 minutes. Add the vinegar and olives and cook until the pan is almost dry, about 2 minutes.
Spread the bottom halves of the buns with the cheese, set a burger on each and top with the shallot-olive relish. Close the burgers and serve.