►Dice butter, transfer to bowl, and refrigerate. Add flour and salt to the bowl of a food processor and pulse a few times to mix. Add butter and shortening. Pulse 8 to 12 times, until butter is in pea-size chunks. With machine running, pour in ice water and pulse until dough begins ball. Dump out on a floured surface and form into a ball. Wrap in plastic wrap and refrigerate at least 30 minutes.
►Heat oven to 400°F. Remove dough from refrigerator and cut in half. Roll each half into a 12-inch circle. Line 2 baking sheets with parchment paper. Transfer each dough circle to a prepared baking sheet.
►Sprinkle cheese evenly over each dough circle, leaving a 2-inch border. Top with onion slices, thyme, tomato slices, salt, and pepper. Fold edges of dough over filling, covering at least 2 inches of filling, pleating as you go. Brush dough with beaten egg.
►Bake galette 40 to 45 minutes, or until crust is brown and tomatoes are soft. Remove from oven, top with chives and flaky salt, and cool on baking sheet for 5 minutes. Transfer to a wire rack and cool at least 10 minutes before serving.