Let's cook.
In a large bowl, whisk together the buckwheat flour, 250 milliliters cold water, 1 large egg, and 2 grams fine sea salt until smooth. Cover the bowl and let the batter rest at room temperature for at least 30 minutes, or up to 2 hours.
Heat a 9-10 inch non-stick pan or crepe pan over medium heat. Melt 5 grams of unsalted butter, swirling to coat the pan.
Pour about 1/2 cup of batter into the hot pan, tilting and swirling quickly to spread it evenly into a thin circle. Cook for 2-3 minutes until the edges are crisp and the bottom is lightly browned.
Flip the galette. Immediately place half of the sliced ham and half of the grated Gruyère cheese onto the center of the galette. Crack one of the remaining large eggs directly onto the cheese.
Cook for another 2-3 minutes, or until the egg white is set and the yolk is still runny (or cooked to your preference).
Once the egg is cooked, fold the edges of the galette towards the center, creating a square or octagonal shape, leaving the egg yolk exposed.
Slide the finished galette onto a plate. Repeat the process with the remaining batter and ingredients, adding another 5 grams of butter to the pan if needed. Serve immediately.
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