In a mixing bowl, whisk together buckwheat flour, water, 1 egg, and salt to make a smooth batter. Let it rest for 10 minutes.
Heat a non-stick skillet or crepe pan over medium heat and lightly grease with butter or oil.
Pour a small ladle of batter into the pan, spreading it thinly to form a large, thin galette.
Cook for 2-3 minutes until edges lift and bottom is golden, then flip carefully.
Place a slice of ham and some grated cheese on one half of the galette. Crack an egg onto the other half, cover, and cook until the egg is set and cheese melted.
Fold the galette in half over the filling, transfer to a plate, and repeat with remaining batter and fillings.
Serve warm.