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For the dough: Combine the flour, sugar and 1/2 teaspoon salt in a food processor and pulse until mixed. Add about a third of the chilled butter and process until thoroughly combined with no visible chunks of butter (the mixture will be slightly yellow). Add the remaining butter and pulse until the mixture resembles a very coarse meal with pea-size bits of butter. Add the vinegar in a single pulse. Gradually add 2 tablespoons ice water through the feed tube, pulsing just until evenly combined. Squeeze a handful of the dough together-it should just hold its shape and be a little crumbly. If still very powdery, pulse again, adding up to 2 tablespoons ice water.
Turn the dough out onto a work surface and pat together into a flat disk about 1/2-inch thick. Wrap tightly in plastic wrap. Refrigerate until firm, at least 1 hour or up to 2 days.
For the galette: Adjust an oven rack to the lower third position and preheat the oven to 400 degrees F.
Toss the yellow squash and zucchini with 3/4 teaspoon salt in a bowl. Let sit for 15 minutes to draw out a good amount of water from the vegetables.
Meanwhile, lay out the bacon slices in a large skillet. Place over medium heat and cook partially, flipping occasionally, to render out some fat but not to brown or crisp up, about 5 minutes. Transfer to a cutting board; let cool slightly, then cut into 1/2-inch pieces. Reserve the bacon fat in the pan.
Pat the squash slices dry; place in a large bowl with the red onion slices and toss with olive oil, 1/2 teaspoon salt and a few grinds of pepper.
Let the dough stand at room temperature for a few minutes until slightly softened. Roll out on a lightly floured surface into a 12- to 13-inch circle, about 1/8-inch thick. Transfer to an ungreased baking sheet. Sprinkle the mozzarella and Parmesan on the circle, leaving empty a 1 1/2-inch border all around. Fan the squash, zucchini and onions in a circle, then scatter the bacon on top. Fold up the edges of the dough, pleating, to overlap the filling. Brush the dough border with the egg wash. Drizzle any bacon fat over the filling.
Bake until golden brown all over and the filling is bubbly, 45 to 50 minutes. Brush the galette with a little vinegar while still warm and serve.