In your food processor, add flour, salt, and baking powder and pulse briefly until combined. Add butter pieces and pulse until crumbly. Add sour cream and pulse briefly until dough comes together.
(If you don't have a food processor, whisk together the flour, salt, and baking powder. Add the butter, working it in with a pastry cutter or 2 knives until crumbly. Leave most of the butter in large, pea-sized pieces. Gently stir in the sour cream; the dough may still be crumbly.)
Turn your dough out onto a floured work surface, and bring it together with a few quick kneads.
Pat the dough into a rough log, and roll it into an 8″ x 10″ rectangle. Dust both sides of the dough with flour, and starting with a shorter end, fold it in three like a business letter.
Flip the dough over, give it a 90° turn on your work surface, and roll it again into an 8″ x 10″ rectangle. Fold it in three again.
Wrap the dough, and chill for at least 30 minutes. Or chill the dough in the fridge overnight and remove 15 minutes before using.
To assemble the galette:
Preheat your oven to 375°F and place a rack on the middle shelf. Line 2 baking sheets with parchment paper.
Break dough into 4 pieces and roll each quarter out to a rough circle approx 1/4-1/2" thick. Transfer dough circles to the parchment-lined baking sheets.
Smooth approx 3 tablespoons of ricotta cheese onto each round, stopping 1 inch from the edge, then sprinkle each with approx 1/4 cup of mozzarella. Fold edges of dough over cheese. Transfer baking sheets to the refrigerator to chill for 15 minutes.
Whisk one egg in a small bowl and use it as an egg wash, gently brushing the edges of each galette.
Bake the galettes for 25-30 minutes, until the pastry is just golden brown.
Sprinkle on slices of proscuitto, slices of basil, and then crack an egg on top of each galette. Bake again for 15-20 minutes, until whites are set and yolks are a bit runny.