Mix everything together until well combined. Let the batter sit in the fridge for 2 hours or overnight.
Heat an 11” cast-iron crepe griddle pan on medium until very evenly hot. For the characteristic craters, the pan needs to be hot enough to immediately punch holes through the batter when poured.
Melt enough butter to coat the pan. Pour 1/2 cup of batter and tilt the pan to cover its whole surface.
Cook approximately 2 1/2 minutes on the first side, then flip and cook for another 1 1/2 minutes. Remove the galette from the heat and let cool until needed for fillings. Repeat with all of the batter, adding butter to the pan as needed to prevent sticking.
For "complète" assembly, melt some more butter and toss in a cooled galette, craters down, and immediately put a slice of the ham in the middle, followed by the grated gruyere to cover the slice. Meanwhile, in a separate skillet cook an egg in some melted butter; when the egg is nearly cooked, gently place it over the gruyere with the yolk centered and fold over the four edges of the galette to leave only the egg visible.
Cover with a lid and keep heating for a minute or so, until the egg is cooked and the bottom of the galette is crisp. Serve immediately.