For the dough: Pulse the flour and salt together in a food processor. Add the butter and pulse about 10 times until the mixture resembles coarse cornmeal with a few bean-size bits of butter in it. Add the egg and pulse 1 to 2 times more; don't let the dough form a mass around the blade. If the dough seems very dry, add up to 1 tablespoon of cold water,
1 teaspoon at a time, pulsing briefly. Remove the blade and bring the dough together by hand. Shape the dough into a disk, wrap it in plastic wrap, and refrigerate at least 1 hour.
For the filling: Halve and core the apple. Cut each half into 8 wedges and put them in a large bowl. Slice the squash and cut the onion into wedges so that both are as thick as the apple wedges and add them to the apples. Add the butter, rosemary, and thyme and toss gently to combine. Season with salt and pepper and toss again.
Preheat the oven to 400 degrees F.
Roll the dough on a lightly floured surface into a 12-inch disk. Transfer the dough to a baking sheet and brush with mustard. Starting 2 inches from the edge, casually alternate pieces of apple, squash, and onion in overlapping circles. If you have extra pieces of one or another, tuck them in where you can or double them up to use all the filling. Fold and pleat the dough over the edge of the filling. Bake until the crust is brown and the apple, squash, and onion are tender and caramelized, about 55 minutes. Scatter the cheese over the filling and bake until melted, about 5 minutes more. Cool the galette briefly on a wire rack. Cut into wedges and serve.