Step 1

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.

Step 2

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.

Step 3

(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.)

Step 4

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.

Step 5

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.)

Step 6

Using your rolling pin and starting on one end of the dough, loosely roll up your pie crust around the rolling pin. Transfer it to the prepared baking sheet and roll the dough back out flat.

Step 7

Spread about 1/3 cup of pesto in the center of the dough, leaving about a 1 1/2- to 2-inch border all around the edge. Lightly drizzle on just a touch of extra-virgin olive oil over the pesto. Place your sliced summer squash, zucchini, and tomatoes in a single layer over the top, arranging them however you want.

Step 8

Gently fold the pastry border over the vegetables, overlapping the edges as much as possible and creating pleats all around the edges. Brush the folded edges of the crust with the egg wash.

Step 9

Bake for 35 to 40 minutes, or until the crust is golden-brown. Let the galette cool for at least 5 to 10 minutes before serving, and garnish with freshly grated Pecorino cheese. Serve the galette with the leftover pesto if desired.