Preheat the oven to 190C. Remove the dough from the refrigerator. Cut dough into 2 - we will be making 2 medium sized galettes.
On a floured work surface, roll out one of the dough into a flat round shape. Don't have to be too perfect here. The key word is "rustic". Transfer the rolled out dough to a baking sheet lined with baking paper.
Scatter half of the grated parmigiano cheese onto the dough, leaving about 2 inch space around the border. Top half of the sautéed mushrooms over the cheese. Fold the edges of the dough border inwards over the mushrooms. Brush the edges of the dough with egg wash. Repeat with the 2nd dough.
Transfer the baking sheet with the 2 unbaked galettes to the refrigerator for 10 minutes. After 10 minutes, remove from the refrigerator and brush egg wash again. Bake in the preheated oven for 15 minutes.
After 15 minutes, take out the galettes from the oven. Use a spoon, push the mushrooms from the middle towards the sides to make a hole for the egg. Crack an egg on top of each galette into the hole and transfer back to the oven and bake for a further 6 minutes. The eggs should look opaque and gooey.
Garnish with sprigs of thyme and sprinkle more grated parmigiano cheese on top. Serve!