Preheat your oven to 475°F (245°C).
Make a 2-inch (or thereabouts) hole in the center of each slice of bread using a cookie or biscuit cutter, an overturned glass, or what have you.
In a large ovenproof skillet over medium heat, melt the butter until it's bubbling but not yet brown. Add the bread and cook until golden brown, maybe 1 to 2 minutes. (Don’t forget to also toss in the little round cut-outs so you have something to dip into the runny yolk.) Flip the bread and carefully crack an egg into each hole. (There may be a little overflow of white onto the bread. As long as you’re not a perfectionist, this is fine.) Quickly sprinkle each slice of bread with 1 tablespoon brown sugar, if desired, being careful to avoid the egg.
Immediately transfer the skillet to the oven to bake. (Don’t try to flip the bread again.) Bake until the egg white sets but the yolk is still runny, 6 to 8 minutes. Begin to check your egg at 4 minutes if you fancy a really runny yolk, as a firm yolk ruins the pleasure of this simple dish.
Lightly sprinkle both bread and egg with fleur de sel. Your egg in a hole is best the moment it emerges from the oven.