To elevate our tart to entrée status, we wanted a crust that was delicate, sturdy, and flavorful, with a filling to match. To increase the flavor of the crust and keep it tender, we swapped out part of the white flour for nutty whole wheat, and we used butter rather than shortening. To punch up its flaky texture and introduce more structure, we gave the crust a series of folds to create numerous interlocking layers.
1. FOR THE DOUGH: Pulse flours, sugar, and salt in food processor until combined, 2 to 3 pulses. Add butter and pulse until butter is cut into pea-sized pieces, about 10 pulses. Transfer mixture to medium bowl.
2. Sprinkle water and vinegar over mixture. With rubber spatula, fold mixture until loose, shaggy mass forms with some dry flour remaining (do not overwork). Transfer mixture to center of large sheet of plastic wrap, press gently into rough 4-inch square, and wrap tightly. Refrigerate for at least 45 minutes.
3. Transfer dough to lightly floured work surface. Roll into 11- by 8- inch rectangle with short side of rectangle parallel to work surface. Using bench scraper, bring bottom third of dough up, then fold upper third over it, folding like business letter into 8-by-4-inch rectangle. Turn dough counterclockwise 90 degrees. Roll out dough again, perpendicular to edge of work surface, into 11-by-8-inch rectangle and fold into thirds. Turn dough 90 degrees counterclockwise and repeat rolling and folding into thirds again. After last fold, fold dough in half to create 4-inch square. Press top of dough gently to seal. Wrap in plastic wrap and refrigerate for at least 45 minutes or up to two days.
4. FOR THE FILLING: Place spinach and 1/4 cup water in large microwave-safe bowl. Cover bowl with large dinner plate (plate should completely cover bowl and not rest on spinach). Microwave on high power until spinach is wilted and decreased in volume by half, 3 to 4 minutes. Using potholders, remove bowl from microwave and keep covered for 1 minute. Carefully remove plate and transfer spinach to colander set in sink. Using back of rubber spatula, gently press spinach against colander to release excess liquid. Transfer spinach to cutting board and roughly chop. Return spinach to colander and press again with rubber spatula; set aside. Add squash to now-empty bowl, cover with plate, and microwave until just tender, about 8 minutes.
5. Meanwhile, heat 1 tablespoon of olive oil in 12-inch skillet over medium heat until shimmering. Add onion and oregano, cover, and cook, stirring frequently, until onions are tender and beginning to brown, 5-7 minutes. Remove from heat, and add onion mixture to squash along with spinach, cheese, crème fraîche, and vinegar, and stir gently to combine. Season to taste with salt and pepper and set aside.
6. TO BAKE: Adjust oven rack to lower middle position, place pizza stone on oven rack, and heat oven to 400 degrees. Remove dough from refrigerator and let stand at room temperature for 15 to 20 minutes. Roll out on generously floured (up to 1/4 cup) work surface to 14-inch circle about 1/8-inch thick. (Trim edges as needed to form rough circle). Transfer dough to parchment-lined rimmed baking sheet. With plastic drinking straw or tip of paring knife, cut five 1/4-inch circles in dough (one at center, and four evenly-spaced midway from center to edge of dough). Brush top of dough with 1 teaspoon olive oil.
7. Spread filling evenly over dough, leaving two-inch border around edge. Drizzle remaining teaspoon olive oil over filling. Carefully grasp one edge of dough and fold up outer 2 inches over filling. Repeat around circumference of tart, overlapping dough every 2 to 3 inches; gently pinch pleated dough to secure, but do not press dough into filling. Brush dough with egg and sprinkle evenly with kosher salt.
8. Lower oven temperature to 375. Bake until crust is deep golden brown and filling is beginning to brown, 35 to 45 minutes. Cool tart on baking sheet on wire rack for 10 minutes. Using offset or wide metal spatula, loosen tart from parchment and carefully slide tart off parchment onto cutting board. Sprinkle with parsley, cut into wedges, and serve.