Prepare the crust. In a medium bowl whisk flour with salt. Incorporate butter using a pastry blender, a fork or a pastry processor, until crumbs are formed. Add water and stir until incorporated.
Divide the dough in 4 pieces and wrap in plastic wrap, shape into disks and refrigerate for at least 30 minutes.
Meanwhile prepare the filling. Heat oil in a large skillet over medium-high heat. Add onion and garlic and cook for about 5 minutes or until softened.
Add mushrooms and cook with the onion for about 10-15 minutes stirring frequently.
Add soy sauce, salt, pepper and thyme and cook another 3 to 5 minutes.
Remove from heat and set aside to cool.
Preheat oven to 400F (200C).
On a lightly floured surface roll each piece of dough into a 8 inch (20cm) circle.
Place each onto a parchment paper lined baking sheet.
Spread cheese over each dough leaving about ½ inch (2cm) around the edge.
Spread mushroom mixture on top of cheese.
Fold dough over the filling. Brush with beaten egg.
Bake for about 20 minutes until slightly golden.
Remove from oven and make a well into the center of each galette.
Crack 4 eggs into 4 small different bowls. If eggs are too large remove some of the whites. Add an egg on each galette and bake for 10 minutes more until eggs are cooked. Garnish with fresh thyme and freshly ground black pepper and enjoy !