In a food processor, combine flour, sugar, and salt. Add cold, cubed butter and, using your fingers, work the butter into the flour mixture. Pulse the food processor until some butter pieces are the size of oat flakes (some will be the size of peas). Add vinegar mixture. Pulse only until liquid is mixed in and all the flour is incorporated in the dough.

On a lightly floured work surface, dump out the dough mixture. It will be moist and shaggy with flecks of butter that haven’t been mixed in. That’s perfect. Gather dough together quickly and, with cool hands, form into a round disk. Wrap in plastic and refrigerate for one hour.

Preheat the oven to 425 degrees Fahrenheit. Line a heavy-duty sheet pan with aluminum or parchment paper and grease with extra virgin olive oil.

Using a small spoon, scoop seeds out of the tomatoes and arrange on prepared sheet, cut sides up. Whisk together olive oil, balsamic vinegar, honey, sugar, salt, and thyme. Pour into and around the rims of each tomato half. Season with additional kosher salt.

Roast until tomatoes are brown at the edges and are mostly collapsed, about 60–70 minutes. Let cool.

Remove dough from fridge and let it sit at room temperature for 20–30 minutes.

On a lightly floured surface, roll dough out into about a 13-inch round. The circle won’t be perfect, which is fine, creating that rustic look. Transfer the dough to a parchment-lined sheet. If you get tears in the dough, they can be patched together with extra dough. When you roll the dough and you can see it start springing back, that means that the butter is warming and the crust shouldn’t be rolled out anymore. (Either add the tomatoes at this point, or put it back in the fridge for another 30 minutes.)

Preheat oven to 400 degrees Fahrenheit.

To assemble the galette, sprinkle middle of dough with three tablespoons Parmesan cheese (leaving a one- and- a- half-inch border where there will be no filling). Add fresh mozzarella slices. Arrange tomatoes over cheese in an overlapping pattern. Sprinkle with two teaspoons Parmesan cheese. Fold edges over tomato filling. Brush dough with egg yolk and sprinkle with remaining Parmesan. Bake until the crust is deeply golden and the bottom is crisp and brown, about 40 minutes. Let cool slightly before slicing.