In a stand mixer, combine: all-purpose flour, instant yeast, salt, and sugar.
Add 2 eggs and 330g cold water. Mix on speed 1 for 1 minute until combined, then increase to speed 4 and knead for 10 minutes.
Add the room temperature softened butter and continue mixing on speed 4 for another 10 minutes, until the dough becomes soft, smooth, and elastic.
Transfer the dough to a lightly greased bowl and cover with plastic wrap. Let it rise until nearly doubled in size, about 1–2 hours at room temperature, or refrigerate overnight if you’d like to prepare the dough ahead of time.
Transfer the dough to a work surface, gently deflate it, and divide it into 8 equal pieces.
Roll each piece into a ball, then flatten into an approximately 8-inch dough disc.
Line a small bowl with the dough sheet, allowing the edges to hang over slightly.
Layer the filling inside with 1 slice of cheese, 1 beef patty, 2 tablespoons of caramelized mushrooms and onions, 2 pickle slices, another beef patty, and 1 more slice of cheese.
Seal the dough tightly around the filling. Pinch the edges together and flip the dough over so the sealed side is on the bottom, forming a smooth ball shape.
Place the closed burger dough balls onto a parchment paper lined baking sheet.
Proof the dough in the oven or in a warm place for 30 minutes. If your oven does not have a proof setting, place a small pot of boiling water next to the baking sheet inside the oven to create a warm and humid environment.
Brush the tops with egg wash and sprinkle sesame seeds on top.
Bake the closed burgers at: 180°C / 350°F on the lower level of the oven, for 15 - 18 minutes Until golden brown.
Serve with your favorite burger sauce, such as a mayo-ketchup sauce, honey mustard sauce, or a creamy mayo Dijon mustard sauce.