Cut 2 x 2-2.5cm thick slices of bread. Remove the crusts and cut each slice of bread into 3 wide fingers. Beat 1 of the eggs in a shallow bowl and season. Dip the bread fingers first into the beaten egg and then into the grated Parmigiano Reggiano. Finish by wrapping a piece of bacon around the middle of each one, like a band.
Warm a large non-stick frying pan over a high heat. Add the butter. When it melts and becomes frothy add the bread fingers, bacon-seam-side down. Cook for 1-2 minutes on each side, or until golden.
Meanwhile, place the remaining eggs into a pan and cover with at least 1cm of cold water. Bring to the boil over a high heat and immediately start timing. Boil for 3 minutes for a very soft runny yolk, or for up to 6 minutes for a very hard egg with completely set yolk. Serve the bread fingers with the boiled eggs, ready for dipping.