Position oven rack in bottom third of oven. Preheat oven to 425°F (220°C). Line a rimmed baking sheet with parchment paper.

On prepared pan, toss together mushrooms, kosher salt, and pepper; spread in a single layer.

Bake until slightly dried, 10 to 15 minutes. Let cool completely on pan. Drizzle vinegar onto mushrooms, and toss to coat. Leave oven on.

In a medium bowl, stir together cream cheese, Emmentaler cheese, and egg yolk until smooth.

On a lightly floured surface, divide All-Butter Piecrust into 4 portions (about 90 grams each). Roll each portion into an 8-inch circle. Spread ¼ cup (70 grams) cream cheese mixture onto each dough circle, leaving a 2-inch border around edges. Top each with 2 tablespoons(20 grams) mushroom mixture. Fold sides of dough over filling, pleating as desired. Place galettes on a parchment paper-lined baking sheet, and refrigerate until firm, about 20 minutes.

In a small bowl, whisk together 1 egg and 1 tablespoon (15 grams) water; brush onto dough.

Bake until crust is lightly browned, about 10 minutes. Crack 1 egg into center of each galette. Bake until crust is golden brown and eggs are just set, 8 to 10 minutes more, or to desired degree of doneness. Let cool on pan for 10 minutes. Garnish with sea salt and chives, if desired. Serve warm.