Make the pastry
In a food processor, pulse the 1 1/4 cups of flour with the salt, pepper and nutmeg. Add the butter and process until coarse crumbs form. Add the water in a slow, steady stream until the pastry comes together. Cover with plastic wrap and refrigerate for about 20 minutes.
Turn out the pastry onto a piece of parchment paper lightly dusted with flour; roll out the pastry to a 14-inch round. Transfer the pastry and parchment to a baking sheet and refrigerate.
Meanwhile, make the almond frangipane
Using a food processor, pulse the almonds with the flour, salt and pepper until finely ground. Add the butter and eggs and process until smooth. Cover and refrigerate.
Make the mushroom filling
In a large skillet, heat the butter. Add the garlic and mushrooms, season with salt and pepper and cook over moderately high heat, until the liquid evaporates and the mushrooms are golden, about 10 minutes. Remove the skillet from the heat and discard the garlic; add three-quarters of the cheese and toss.
Preheat the oven to 400º. Remove the pastry from the refrigerator. With the pastry still on the baking sheet, spread the frangipane on the pastry to within 2 inches of the edge. Top the frangipane with the mushroom filling. Fold the edge of the dough up and over the filling to create a 2-inch border; brush with the egg wash and sprinkle with salt and pepper. Scatter the remaining cheese all over the galette and bake for about 50 minutes, until the crust is golden. Transfer the pan to a rack and let the galette cool. Serve warm or at room temperature.