Prepare the dough: Combine the flour, sugar, 1 teaspoon salt, and olive oil in a large mixing bowl. Toss the butter cubes in the flour to coat, then gently press them with your fingertips to flatten. There should be large, thin pieces of butter visible throughout the mixture.

Add 4 tablespoons ice water and toss to combine, adding additional water one teaspoon at a time until the dough just comes together when pressed with no dry streaks of flour. Wrap the dough in plastic wrap and refrigerate for at least 1 hour.

Prepare the tomatoes: While the dough rests, halve, core, and thinly slice the tomatoes. Discard the cores. Lay the tomato slices on a plate and season with sea salt and black pepper. Set aside to allow the tomatoes to release their liquid at room temperature for up to 1 hour.

Preheat the oven to 425°F.

Lay the chilled dough on a sheet of parchment paper on a flat surface and roll it into a rough circle about ¼ inch thick. Lift the tomatoes from the residual juice and pile them onto the dough in layers with the basil, pecorino cheese, and aged balsamic vinegar. Leave a 1 ½ inch border around the edge of the dough circle. Discard the excess tomato juice.

Pull the edges of the dough up over the tomatoes to form the galette, leaving the tomato filling exposed in the center. If desired, grate additional pecorino cheese over the crust. Lift the parchment paper carefully to transfer the galette to the pan.

Bake for 30 minutes at 425°F and check for doneness. If the crust needs to continue browning, lower the oven to 375°F and continue baking until golden brown, up to 15 additional minutes depending on your oven.

Remove the galette from the oven and rest on the baking sheet for at least 30 minutes before slicing and serving.