Put flour, sugar, baking powder, and salt into a food processor and pulse until combined.

Add the cold butter and pulse just a few times until the pieces of butter are no bigger than a pea.

Add the egg and vanilla extract, and pulse several more times until the dough begins to clump together and form a ball.

Generously butter the inside of a springform pan and place parchment paper at the bottom, buttering the top of the parchment as well.

Press the dough evenly onto the bottom of the pan (about 1/4 inch thick) and about 2 inches up the sides.

Chill the crust in the refrigerator for at least 10 minutes.

Preheat the oven to 375°F with the oven rack in the middle position.

Grind 2 tablespoons sugar with flour and tapioca until powdery, then transfer to a large bowl and stir in the remaining sugar.

Add the peaches, blueberries, and lemon juice to the bowl, gently tossing to coat.

Fill the chilled pastry crust with the peach and blueberry filling. Cover loosely with foil.

Bake at 375°F until the filling is bubbly and the crust is golden, about 1 3/4 hours.

Remove the cake from the oven and let it cool on a rack for 20 minutes. Run a blunt knife around the inside edge of the pan to release the crust, then carefully remove the sides.

Cool the cake to barely warm or room temperature before cutting into thick wedges.