To make the pesto
Before working on the pesto, begin work on the galette pastry. See below
Add all the ingredients together into the food processor and process until it forms a smooth paste.
To make the galette
In a food processor, add the flour, herbs, salt, parmesan or manchego cheese, and butter, and pulse for 5 seconds to blend it
Mix the cold water and vinegar in a glass. Slowly add the water, little by little, and pulse until the pastry starts to come together. When you touch it, you should be able to press and mold it. It should have the texture of wet sand
Transfer it to your work surface and roll it into a ball and flatten it into a disc. Plastic wrap it and refrigerate it until it is chilled. While the dough is chilling, prepare the pesto
Once chilled, sprinkle some flour on your work surface and roll out the dough to a 14 inch oval. Make sure you work quickly so that the dough remains cool. Preheat the oven to 375 degrees
Spread and smooth out the pesto across the surface of the dough, leaving a little more than an inch for the crust
Layer the tomato and fresh mozzarella by placing alternate slices on the pesto. Drizzle more pesto on top
Fold the crust part of the dough up and over sort of forming a tucked wall. Mix the yolks and the milk or heavy cream to make an egg wash. Using a brush, spread the egg wash across the outside. Add in the middle rack of the oven and bake for about an hour or until the pastry has browned and crisp
Let cool before serving. Garnish with some more chopped, fresh basil