Carefully mix the meats, steak sauce, egg yolks, salt, and pepper with the tines of a fork, but do not mash it. Lightly form each hamburger and lightly press into a patty shape. Make an indentation and put a thin slice of butter inside each hamburger, making sure the butter is entirely encased in the meat.
Heat a grill, broiler, or saute pan and cook the hamburgers for 3 to 5 minutes on each side until almost done. Remove to a plate and cover with aluminum foil. Allow the hamburgers to rest for 5 minutes and serve immediately on buns.