In a bowl, mix the flour, Parmesan, and salt. Add the cold butter and work it in with your fingers until the texture is sandy.
Add the buttermilk, then 1–2 tablespoons of cold water as needed, just until the dough comes together. Shape into a disc, wrap, and chill for at least 1 hour.
Halve the cherry tomatoes, sprinkle with salt, let them sit for a bit, then pat them dry with a paper towel.
Preheat the oven to 190°C (375°F).
In a bowl, combine the ricotta, Parmesan, and feta.
Roll the dough out (about ½ cm thick) on a lightly floured parchment paper.
Spread the cheese in the center, leaving a 3 cm border. Arrange the tomatoes on top and drizzle with olive oil.
Fold the edges of the dough over the filling. Brush the crust with beaten egg. Bake for 30–40 minutes, or until golden.