To make the crust: In a food processor, add the flour, sugar and salt, and pulse 3 to 5 times, until combined.
Add the chilled butter and pulse until the mixture resembles coarse crumbs.
Add the egg yolk, then slowly add the cold water while processing the mixture. Process until the dough just starts to form a ball.
Transfer the dough to a lightly floured surface, and shape it into a disc. Wrap the disc tightly in plastic wrap and refrigerate for at least 1 hour.
To make the filling: Heat the oven to 375 F.
Trim off the woody ends of the asparagus. Thinly slice each spear on the bias, about ¼-inch long.
In a large skillet, heat the butter over medium-high heat until it’s foaming. Add the asparagus and sauté for 2 minutes. Add the leeks, garlic, crushed red pepper and thyme, and sauté for 3 to 5 minutes, or until everything is just softened.
To assemble the galette: On a lightly floured piece of parchment, roll the dough into a circle, about 14 inches in diameter and ¼-inch thick. Transfer the dough and parchment onto a baking sheet. The dough should hang slightly over the long edges of the baking sheet.
Sprinkle the Gruyère and Parmesan over the dough, leaving a 2-inch border around the edge. Spread the sautéed vegetables on top of the cheese in an even layer.
Fold the edge of the dough over the filling, pleating the dough as you go around. Brush the dough with the beaten egg.
Bake for 35 to 40 minutes, or until the crust is golden brown.
Let the galette cool for 5 minutes. Garnish with additional Parmesan, then slice and serve warm or at room temperature.