Prepare the Batter: In a large bowl, add 1 cup buckwheat flour (125 grams). Make a well in the center, crack in 1/2 egg, and sprinkle 1/2 teaspoon coarse salt (2.5 grams) around the edge. Using a whisk, start mixing from the center outward, gradually adding 1 cup cold water (250 ml). The batter should be thick but pourable. Add 2 teaspoons salted butter (10 grams), cut into small pieces. Cover the bowl and refrigerate the batter for 2 to 3 hours.

Adjust the Consistency: After resting, whisk the batter again to incorporate any water on the surface. Make sure the batter is smooth, thick, and pourable. Set aside for cooking the galettes.

Assembling the Galettes: Melt 1 teaspoon salted butter (about 5 grams) in a skillet over medium heat. Place a galette in the heated skillet, add half a slice or some shredded ham in the center, and generously sprinkle shredded Gruyère cheese over the ham.

Heating the Galettes: Cook until the cheese starts to melt. Crack 1 egg into the center of the galette and gently spread the egg white over the cheese to help it set. Continue cooking until the egg white turns opaque.

Folding the Sides: Using a spatula, carefully fold the four sides of the galette toward the center to form a square. Press gently with the spatula to seal the edges.

Finishing: Keep the assembled galette on medium heat for about 1 more minute, until the egg yolk in the center is still slightly runny. Cover the skillet with a lid if needed to help cook the egg.