Combine eggs, milk, and thyme in a shallow bowl and whisk until blended. Butter one side of the bread, dip about half the slices in the egg mixture, and use to cover the bottom of a greased 9x13-inch baking dish. (If necessary, cut bread into triangles to fit pan.) Sprinkle with half the cheese and half the meat. Top with the remaining buttered bread, then the remaining cheese and meat. Pour remaining egg mixture over the top, cover, and refrigerate for 24 hours. Bake uncovered in a preheated 350 degrees F oven for 1 hour. Let sit 15 minutes before serving.