Potato, Parmesan, and Caramelized Onion Galette

For the dough:
1 stick cold unsalted butter
1 1/3 cups flour
1/4 teaspoon salt
1 pinch sugar
1/4 cup grated Parmesan
1/3 cup ice water (only add as much as needed)
 
For the caramelized onions:
2 tablespoons olive oil
1 white onion, thinly sliced
1/4 teaspoon sugar
1 pinch salt, or to taste
3 cloves garlic, minced
1/3 cup dry white wine
1 tablespoon chopped sage
 
For the filling:
Caramelized onions (above)
1 pound Yukon Gold potatoes, sliced thin
1 teaspoon salt, and pepper to taste
1 1/4 cups grated Parmesan or Asiago
5 sprigs thyme, leaves removed from stems
3 tablespoons butter, melted
Zest from one small lemon
1 egg, beaten
 
Make the dough:
(Can be made ahead and refrigerated for a day or two.)

Grate the butter into a small bowl and place in the freezer for about 8 to 10 minutes. In a large bowl, mix flour, salt, sugar, and cheese. When the butter is nice and cold, incorporate it into the flour mixture using a pastry cutter, your fingers, or a food processor. When the mixture begins to clump together, slowly add water, only adding as much as needed to form a ball – start with 1/4 cup and increase as you mix. You don’t want a wet dough. On a floured surface, shape ball into a disc and wrap tightly with plastic wrap. Refrigerate for an hour before using.
 
Caramelize the onions:
Heat olive oil in a medium saucepan over medium heat. Add onions and stir frequently so they brown but not burn – adjust the heat as necessary. After about 20 minutes, add sugar and season with a bit of salt. Continue cooking and stirring until onions are a deep golden brown. This can take about 45 minutes or longer. Don’t wander away and leave them unattended, as they can stick and burn. When the onions have acquired a nice deep golden tan, add in the garlic and cook, stirring, for a few more minutes.  Then add wine and sage and cook for about 2 to 3 minutes, until the liquid has evaporated. Set onions aside to cool before assembling the galette.
 
Assemble the galette:
Heat oven to 400°F.
 
Lay out a large sheet of parchment paper on your work surface. Remove dough from refrigerator and, with a lightly floured rolling pin, roll out dough into a somewhat round shape, about 1/8 inch thick. Place about half of the onions in the center of the dough, followed by half of the potato slices. Leave about a 1 1/2- to 2-inch border. Lightly season with salt and pepper and top with half of the cheese and thyme. Drizzle with half of the butter. Follow with the rest of the potatoes, more salt and pepper, the remaining onions, cheese and thyme, and lemon zest. Fold the edges of the dough over, brush with egg wash, and bake for about 45 to 50 minutes, until golden brown. Let cool for about 5 to 10 minutes before serving.

From Epicurious