Pulse flour, sugar, and salt in food processor to combine. Add butter and process until largest pieces of butter are pea-size. Transfer to a large bowl.
Combine vodka, vinegar and 2 Tbsp. ice water (or 1/4 cup ice water if not using vodka) in a small bowl and drizzle over flour mixture, mixing with a fork to combine. Mix until shaggy pieces form, then knead in bowl a couple of times with your hands to bring together (dough will look very clumpy, dry, with loose bits). Transfer large clumps of dough to work surface, drizzle 2 teaspoons ice water over remaining flour mixture in bowl and knead again to bring it together. Place on top of dough on work surface. Press into a single mass, incorporating dry bits, then pat down to make a 3/4"-thick square. Using a bench scraper or knife, divide dough into 4 pieces. Stack pieces on top of one another, placing any unincorporated dry bits in between layers, and press down to combine. Form dough into a 3/4"-thick square and repeat process 3 more times. Flatten into 3/4" rectangle and wrap tightly in plastic. Chill at least 2 hours.
Place a rack in the lowest part of the oven; preheat to 375°. Let dough sit at room temperature 5 minutes to soften. Roll out dough on a lightly floured surface to ⅛" thick and about 18 x 13”. It’s ok if the edges are shaggy and it’s not quite rectangular! Place on a parchment-lined rimmed baking sheet and chill while you prepare the toppings.
Toppings and Assembly:
Toss mushrooms, oil, half of garlic, 1 1/2 teaspoons salt and pinch of pepper in a large bowl until completely coated. Stir ricotta, parmesan, chopped thyme leaves, 3/4 teaspoons kosher salt, 1/2 teaspoons pepper and remaining garlic in a small bowl until combined.
Sprinkle mozzarella evenly over top of cold pastry. Dollop ricotta evenly over top. Arrange scallions next and spread green ends out so they fan and overlap across the galette. Top with mushrooms, fresnos and thyme sprigs. Bake on lowest rack until the crust is golden brown and the cheese is brown in spots. 50-65 minutes. Let cool 30 minutes and remove thyme sprigs before serving.
Do Ahead: Dough can be made 5 days ahead. Keep chilled, or freeze up to 1 month.