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Preparation Pulse 2 cups flour and 1 1/4 tsp. kosher salt in a food processor to combine. Add butter and pulse until mixture resembles coarse meal with a few pea-size pieces of butter remaining.
Transfer mixture to a large bowl; drizzle with vinegar and 1/4 cup ice water. Mix with a fork, adding more ice water by the tablespoonful as needed, just until a shaggy dough comes together. Turn out onto a work surface and lightly knead until no dry spots remain (be careful not to overwork). Pat into a disk and wrap in plastic. Chill at least 2 hours.
Preheat oven to 400°F. Gently toss tomatoes, garlic, and remaining 1 tsp. kosher salt in a large bowl. Let sit 5 minutes (tomatoes will start releasing some liquid). Drain tomato mixture and transfer to paper towels.
Unwrap dough and roll out on a lightly floured sheet of parchment paper to a 14" round about 1/8" thick. Transfer on parchment to a baking sheet. Scatter cheese over dough, leaving a 1 1/2" border. Arrange tomatoes and garlic over cheese. Bring edges of dough up and over filling, overlapping as needed to create about a 1 1/2" border; brush dough with egg. Sprinkle tomatoes with sea salt and pepper. Chill in freezer 10 minutes.
Bake galette, rotating once, until crust is golden brown and cooked through, 55–65 minutes. Let cool slightly on baking sheet. Finely grate zest from lemon over galette; sprinkle with chives.
Do Ahead: Dough can be made 2 days ahead. Keep chilled.