Whisk the egg and milk in a small bowl and set aside.
Add crumbled goat cheese to the cooled mushroom mixture.
Adjust the oven rack to lower middle position, place baking stone on rack and heat the oven to 400 F.
Remove the dough from the the refrigerator and let stand at room temperature for 15 to 20 minutes.
Roll out on generously floured counter to 14 inch, around 1/8 inch thick circle.
Transfer the rolled dough to parchment paper lined baking sheet. With a tip of paring knife cut five 1/4 inch concentric cricles in the dough, starting from 1 inch away from the center.
Brush the dough with olive oil and spread the filling evenly, leaving 2 inch border around the edge.
Grasp 1 edge of the dough and fold up outer 2 inches over the filling. Repeat around the circumference of the tart, overlapping dough every 2 to 3 inches; gently pinch the pleated dough to secure.
Brush the dough with egg and milk wash and sprinkle some kosher salt if desired.
Lower the oven temperature to 375 F. Bake until the crust is deep golden brown and the filling is starting to look brown, around 40 minutes. Rotate the sheet halfway through baking.
Let the galette cool on a wire rack for 10 minutes before removing it from the baking sheet.
Garnish with fresh thyme leaves.