For the crust: In a food processor fitted with a steel blade, or in a large bowl, pulse or mix flour, sugar, and salt. In a liquid measuring cup, lightly beat the egg, then add just enough cream to get to 1/3 cup. Lightly whisk the egg and cream together.
Add cold butter to the flour mixture and pulse or use a pastry cutter or your fingers to break up the butter. You need chickpea-size chunks of butter. Drizzle the egg mixture (up to 1/4 cup) over the dough and pulse or stir until it just starts to come together but is still mostly large crumbs. Mix in basil, lemon juice and zest.
Put the dough on the lightly floured counter and form it until it comes together. Flatten into a disk, wrap in plastic and chill for 2 hours, or up to 3 days (I refrigerated mine overnight). Heat the oven to 400 degrees. Roll the dough out into a 12-inch round (it can be ragged). Transfer to a rimmed baking sheet lined with parchment paper and chill while preparing the filling.
For the cherry filling: Toss together the cherries, all but a tablespoon of sugar, the salt, the lemon juice and zest, and the cornstarch. Use more cornstarch for juicy stone fruit and less for blueberries, raspberries and figs. Pile fruit on the dough circle, leaving one 1/2-inch border. Gently fold the pastry over the fruit, pleating to hold it in (sloppy is fine). Brush pastry generously with leftover egg and cream mixture. Sprinkle remaining sugar on the crust.
Bake for 35 to 45 minutes, until the filling, bubbles up vigorously, and the crust is golden. Cool for at least 20 minutes on wire rack. Serve warm or at room temperature.