Make the Galette Batter: Melt 1 tablespoon of the butter in a small pot over medium heat.
Combine 2 cups of buckwheat flour, 1 tablespoon muscovado sugar and ¼ teaspoon sea salt in a large bowl.
Whisk 1 egg in a large bowl, then add 4 cups of milk. Blend the liquid into the dry ingredients. Next, whisk in 1 tablespoon of melted butter until smooth.
Store covered in the fridge until ready to use. The batter will need to chill for at least 8 hours.
Make the Melted Onions: Melt 2 tablespoons unsalted butter in a large saucepan. Cook 1 pound of thinly sliced vidalia onions, covered, on low to medium heat until translucent, stirring frequently, about 30 minutes.
Add 1.5 tablespoons of sweet rice flour, stirring until it’s combined. Then whisk in 1 cup of milk and 2 tablespoons of cream. Grate a pinch of fresh nutmeg into the sauce and let it come to a simmer, whisking frequently and making sure to scrape the bottom of the pan so the sauce doesn’t scorch. As soon as the sauce comes to a rapid simmer the heat has activated the starch, and this causes it to thicken. Remove it from the heat. Season with sea salt and freshly cracked pepper to taste and chill until ready to use.
When you are ready to make the galettes, melt a small pat of butter on medium heat in a non-stick crepe or omelet pan.
Once the pan is hot, ladle 2 ounces of batter into the pan. Use a quick rotating motion to make the thin batter coat the bottom of the pan, or use a crepe spatula to spread the batter evenly.
Cook for 15 seconds and then using a spatula and your hands, flip the galette on its other side. Add a slice of cheese and top with a tablespoon of melted onions. Fold the galette in half and transfer to a plate. Continue making the rest of your galettes.
Cook your eggs according to your desired outcome, then place an egg on top of each galette. Toss the arugula with lemon juice and olive oil, place a small amount on the galette and enjoy immediately.