Add the 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.
Press dough into pan:
Generously butter the inside of a 9 or 9 1/2-inch springform pan. Place a round of parchment paper at the bottom of the pan and butter 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 of the pan.
Put into the refrigerator to chill for at least 10 minutes.
Preheat the oven to 375°F:
Put oven rack in middle position.
Make the filling:
Grind 2 tablespoons sugar with flour and tapioca in a spice grinder until tapioca is powdery. (If using cornstarch instead of instant tapioca, you do not need to grind.)
Transfer to a large bowl and stir in remaining sugar. Add peaches, blueberries, and lemon juice and gently toss to coat.
Bake:
Fill pastry crust with peach and blueberry filling. Cover loosely with foil. Bake at 375°F until the filling is bubbly in the center and crust is golden, about 1 3/4 hours.
Cool and remove from pan:
Remove cake from oven and set on a rack to cool, uncovered for 20 minutes. Run a blunt knife around the inside edge of the pan to help release the crust from the sides of the pan. Then carefully remove the side of pan.
Cool cake to barely warm or room temperature Cut into thick wedges before serving.
Note that although the cake will come out of the oven somewhat liquidy. The liquid should gel nicely as the cake cools.
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