For the crust:
1 1/4 cups all-purpose flour
3/4 teaspoon sea salt
1 stick (1/2 cup) very cold unsalted butter; cubed
1/4 cup ice-cold water
1 large egg yolk + 1/2 teaspoon water; slightly beaten (for egg wash)
For the pesto:
1 cup packed fresh basil leaves
1 1/2 cups packed fresh arugula
2 to 3 cloves fresh garlic
1/4 cup pine nuts
1 cup extra-virgin olive oil
Kosher salt and freshly ground black pepper, to taste
1/2 cup freshly grated Pecorino cheese
For the galette filling:
1 teaspoon extra-virgin olive oil + more for drizzling
1 medium-sized yellow summer squash, sliced thin crosswise
1 medium- to large-sized zucchini, sliced thin crosswise
2 small tomatoes, sliced thin crosswise
Freshly grated Pecorino cheese
To make the crust: Combine the flour, salt, and cold, cubed butter in the bowl of a food processor, and process for about 5 seconds. Sprinkle the ice water over the flour mixture and pulse until the pastry just begins to come together, for about 10 seconds; you should still see small pieces of butter in it.
Transfer the pastry dough to a lightly floured work surface, gather it together to form a disk or mound, wrap with plastic wrap, and refrigerate for at least 30 minutes.
(For step-by-step instructions, see How To Make a Pie from Start to Finish; the pie crust can be made several days ahead and kept refrigerated until ready to make the galette.)
To make the pesto: Clean out the food processor and then combine the basil, arugula, garlic, and pine nuts in the bowl. Pulse until the ingredients are coarsely chopped. With the food processor running, slowly pour the olive oil into the bowl through the feed tube, and process until the pesto is thoroughly pureed. Season with salt and pepper and pulse a few more times. Add the Pecorino cheese and puree until the cheese is thoroughly combined. Transfer the pesto to a jar with a lid or covered bowl until ready to use. The pesto can be made several days ahead and kept refrigerated.
To assemble the galette: Preheat the oven to 375°F and line a baking sheet with parchment paper or a Silpat. Remove the disk of pie dough from the refrigerator and place it on a lightly floured work surface. Dust some flour on the rolling pin, and roll out the dough to a large circle, about 12 inches in diameter. (If your dough was refrigerated for more than half an hour, wait about 20 minutes before rolling.)
Using your rolling pin and starting on one en