Make the Crust: Whisk the flour and salt together. Add the cold butter and cut it into the flour mixture using a pastry cutter or two forks, until the mixture resembles coarse meal and just a few pea-sized chunks of butter remain. Add four tablespoons of the ice water and continue mixing. Add water, one tablespoon at a time, until the dough just starts to come together. Using your hands, gather the dough together and press into a ball. If the mixture doesn’t hold together in a ball, add another tablespoon of ice water and try again. Transfer the bowl and the ball of dough to the refrigerator while you prepare the pesto.
Preheat the oven to 375°
Make the Pesto: Process the basil in a food processor until it is coarsely chopped. Add the pistachios and garlic and process until the nuts are very small, stopping to scrape down the sides as needed. With the motor running, drizzle in the olive oil and process until you have a thick paste. Set aside.
Assemble the Galette: Roll the dough out between two sheets of parchment paper into a 12-inch round, about 1/8-inch thick. Remove the top piece of parchment and use your fingers to press the edges together slightly to make a smoother edge. Spread the pesto over the dough leaving a little over an inch border. Sprinkle 3 tablespoons of Parmesan evenly over the pesto. Layer the tomato slices on top of the Parmesan-covered pesto, overlapping them just slightly if necessary to fit them all. Gently fold the border over the filled portion by holding onto the parchment and pulling the dough up and over, overlapping the dough as needed. Use your fingers to patch together any portions that break. Brush the crust with the beaten egg. Sprinkle the entire galette with a little salt, fresh ground pepper, and the remaining tablespoon of Parmesan cheese.
Carefully transfer the prepared galette and the bottom sheet of parchment to a baking sheet. Place the baking sheet in the preheated oven and bake for 30-35 minutes, rotating the baking sheet once about halfway through cooking, until the galette is golden brown. Transfer to a cooling rack and allow galette to cool for 10 minutes before slicing and serving.