Assemble your sandwich, then butter one side of the bread. Separate the egg white and yolk into two bowls. Place the sandwich, buttered-side-down, in a pan over medium heat and grill until golden brown, then remove.
Butter the uncooked side of the sandwich. Using a cookie cutter or a sharp knife, cut out the center. Lower the heat slightly, then return both pieces of the sandwich to the pan, uncooked-side-down. Add just enough egg white to fill the bottom of the empty sandwich center, then carefully slip the egg yolk into the center.
Cover the pan (this will help the egg cook quicker) and let everything cook for a couple minutes, checking regularly after one minute to make sure nothing is burning. Once the egg appears to be cooked, immediately remove the sandwich and serve!