Make the crust: In a food processor, place 1 cup of the flour and the salt. Pulse to combine. Spread the butter pats evenly on top of the flour mixture. Pulse until the butter is well combined with the flour and the dough starts to form clumps. There should be no dry flour remaining.
Spread the mixture evenly in the bowl of the food processor and sprinkle the remaining ½ cup flour on top. Briefly pulse until the dough becomes crumbly.
Transfer the dough to a large bowl and sprinkle the water and vinegar on top. Use a spatula to fold and press the dough until it comes together into a ball. Form the ball into a 1-inch-thick disk. Wrap tightly in plastic wrap and chill for at least 2 hours, or up to 2 days.
Make the filling: Place the tomato slices in a large bowl and gently toss with the salt. Set aside for 30 minutes to draw excess moisture out of the tomatoes. Drain and transfer to paper towels or kitchen towels to soak up additional moisture while you assemble the galette.
In a medium bowl, stir together the goat cheese, olive oil, chives, thyme, garlic, lemon zest, and several grinds of pepper until creamy.
Assemble the galette: Preheat the oven to 400°F and line a large baking sheet with parchment paper.
In a small bowl, whisk together the egg and water.
Place the chilled dough on a lightly floured surface. Dust the top with flour and use a rolling pin to roll it out into a 12-inch circle about ⅛-inch thick. As you work, rotate the dough a quarter turn every few rolls to prevent it from sticking to your work surface. I also like to dust the top of the dough with flour and flip it over a few times during the rolling process.
To transfer the rolled-out dough to the prepared baking sheet, roll the dough over your rolling pin and unroll it onto the baking sheet. Spread the goat cheese mixture in the center of the dough, leaving a 1-inch border all around. Arrange the tomato slices on top and fold the edges of the dough over the edges of the tomatoes, leaving the center of the filling exposed. Brush the crust with the egg wash.
Bake for 40 to 45 minutes, or until the crust is golden brown. Allow to cool for 10 minutes on the baking sheet. Garnish with fresh basil and sprinkle with flaky sea salt. Slice and serve.