Add the flour, salt, basil and oregano to a food processor and pulse just until combined. In a small bowl, whisk together the egg, vinegar and water. Add the cold butter pieces and parmesan cheese into the food processor and pulse until small coarse crumbs remain. Sprinkle the water/egg mixture over the flour and pulse again until the dough comes together.
Remove the dough and wrap it in plastic wrap. Refrigerate one dough for 30 minutes.
Preheat the oven to 400 degrees F.
Line a baking sheet with a paper towel or kitchen towels. Place the sliced tomatoes on the towel in a single layer. Sprinkle all over with salt and let sit for 10 minutes. After 10 minutes, gently pat with a paper towel to remove the excess liquid.
In a bowl, stir together the mayo, garlic, cheddar, parmesan and pesto. Add a pinch of salt and pepper. Stir until combined.
Remove the crust from the fridge. Roll the crust into a… “rustic” shape – no shape necessary really – until it is about ¼ inch thick. Place the dough on a parchment-lined baking sheet. Spread the cheese filling on the dough, leaving a 2 to 3 inch border at the edges.
Once the filling is spread on the dough, top it with the sliced tomatoes. Add a few twists of black pepper (they already have salt!) and fold up the edges of the dough. Brush the dough with the egg wash.
Bake the galette until the crust is golden, about 40 to 45 minutes. Remove and top with fresh basil and parmesan. Let sit for 10 to 15 minutes before slicing. Slice and serve!