Pulse flour, butter, and salt in a food processor until mixture resembles coarse meal with a few peanut-sized chunks of butter. Turn food processor on again and quickly drizzle 4 tablespoons of the ice water through the feed tube, stopping before the dough comes together. Pinch some crumbs between your fingers; if they don’t stick together, add a bit more water and whirl briefly. Turn the mixture out onto a large piece of plastic wrap. Use the plastic wrap to help you bring the dough together; fold the edges of the dough toward the center and press down until it holds together, taking care not to overwork the dough. Press the dough into a flat disk about 5 inches in diameter. Chill dough, wrapped in plastic wrap, for 30 minutes.
In the meantime, preheat oven to 375° F and prepare the filling ingredients. Slice tomatoes into 1/4 to 1/2-inch thick rounds. Set tomatoes aside on a plate to release some of their juices. If they're especially juicy, you may want to set them on a paper towel as well. Turn them at least once as you're preparing other ingredients.
Roll out dough into a 13-inch round on a lightly floured surface. Transfer to a parchment-lined baking sheet. Scatter the cheeses evenly over the dough, leaving a 1 1/2 to 2-inch border around the edges. Sprinkle minced shallots over the cheese and arrange the sliced tomatoes on top, fitting them together as tightly as possible without overlapping. Drizzle olive oil over the tomatoes and scatter thyme and salt on top.
Fold dough in on itself to cover outer rim of filling, pleating dough as necessary. Brush pastry with beaten egg.
Bake galette on the middle rack until the crust is golden and the filling is bubbling, about 30 to 35 minutes. Place baking sheet on a rack and cool for at least 10 minutes before serving.