To make the topping:
Preheat the oven to 400°F. Place a rack in the lower third of the oven. Do not place in the lowest slot or the bottom of the galette will burn before the topping is done.

In a medium skillet, heat the olive oil over medium-high heat. Add the onions and squash and sauté for about 2 minutes, stirring constantly. Add the thyme, salt, and pepper and continue to cook until the onions and squash are translucent but not brown, about 5 minutes. They should be slightly underdone. Remove from the heat and set aside.

To assemble:
Remove the dough from the refrigerator, and on a well-floured surface or on a floured piece of parchment, roll it into a circle 8 to 9 inches in diameter and 1/8 inch thick. Place the dough in a shallow baking or pizza pan. (Do not use a rimless baking sheet, or you'll likely end up with melted butter on your oven floor.)

Spoon the onion-squash mixture onto the center of the dough, leaving a 1 1/2-inch border. Top with the cheese. Next, fold the 1 1/2 inch border of the galette toward the center to encase part of the filling, crimping the edges a little as you go. You should end up with a "window" of filling about 5 inches in diameter, with the crust overlapping the edges of the filling. Brush the folded-over edges with the milk or cream.

Bake until the border is golden and the cheese is bubbly and golden brown, 25 to 30 minutes. Serve immediately.