Slice the goat cheese into 12 (1/2-inch-thick) slices. Dip each slice into the egg wash and then into the bread crumbs, being sure the cheese is thoroughly coated. Place the slices on a baking rack set on a sheet pan and chill for at least 15 minutes.
Heat the oil and butter in a medium saute pan over medium-high heat. Add the goat cheese slices and cook on both sides until browned on the outside but not melted inside.