Preheat oven to 300F° to be able to keep the French toast warm while finishing the Egg In the Hole pieces.
Using a 3" cookie cutter, biscuit cutter, or glass (whatever size will cut into the bread but still leave enough of the bread around the hole to stay together), cut a hole in the center of two slices of bread. Save the piece of bread you cut out if desired, especially if using a heart-shaped cutter. Set aside.
In a medium bowl, add one egg. Sprinkle the egg with the 1 tablespoon of flour and 1 tablespoon of sugar and whisk with a fork to combine. Add the two cups milk, ½ teaspoon black pepper, ½ teaspoon cayenne, and dash of salt. Whisk very well to fully incorporate the egg and milk.
Heat the griddle to 325F°-350F° depending on how hot your griddle cooks. Melt a generous amount of butter on the griddle. Working quickly, first dip the bread that does not have a hole in it into the batter, covering both sides, and place on the griddle. Now dip the bread pieces for the egg in a hole into the batter, being careful with these pieces so the weight of the batter doesn't break the bread. Place these pieces on the griddle also. Dip the leftover shapes from your cutouts, if using, and place on griddle. Let all the bread griddle for a couple minutes until you can tell they're browned but not burning.
Using a good-sized spatula or pancake turner, flip all the bread. I quickly add a little more butter to the griddle as I'm flipping the toast for more good browning, flavor, and to ensure nothing sticks. See Note 1 for how to easily do this. At least add more butter to the griddle for the pieces that will get the eggs.
To ensure egg shells don't get into your eggs that are going in the holes, break one egg into a small bowl and then gently tip the egg into the bread with the hole. Repeat with the second egg into the other piece of cut-out bread.
By now, the regular slices of French Toast and extra shaped pieces should be well toasted on the second side. Remove them from the griddle and place on a baking sheet covered with a wire cooling rack and put into the oven to keep hot while the eggs finish.
Once you see the bottom of the eggs becoming white, about 3 minutes, carefully but quickly flip the egg-in-the-hole bread pieces on the griddle to finish cooking the eggs to your desired doneness. About two minutes on the second side for over-easy, depending on your griddle.
Remove the French toast pieces from the oven to individual plates, two per person, then top each plate with an egg-in-the-hole piece of French toast. Serve topped with warm syrup and berries.