Make the tart dough: In a large bowl, stir the flour and salt until blended. With both hands, work the butter into the flour until the mixture is fine-textured and no large bits of butter remain. Sprinkle the vinegar over the dry ingredients, then sprinkle the ice water, a little at a time, incorporating it with your hands and adding more water just until the ingredients come together into a dough. You may not need all the water. Knead the dough gently until it is smooth. Shape into a 1-inch-thick round, wrap tightly in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 375ºF. On a lightly floured surface, roll out the dough into a round large enough to fit a 9-inch tart pan with a removable bottom. Carefully transfer the round to the tart pan and press into place. Trim off the overhang even with the rim by rolling the rolling pin across the rim. Put a piece of aluminum foil in the pan to cover the bottom and sides and fill with pie weights or dried beans. Bake for 20 minutes, then remove the weights and foil and prick the shell in several places with a fork. Return to the oven and bake until lightly colored, about 5 minutes longer. Let cool before filling.

Make the filling: In a bowl, whisk together the ricotta, egg yolks, Parmesan, tomato oil, generous pinch of salt, and the pepper. Whisk the egg whites with a pinch of salt to firm peaks, then fold gently into the ricotta base. Pour the filling into the cooled tart shell, spreading it evenly. Arrange the tomato quarters, cut side up, in 2 concentric circles, with 16 quarters on the outer circle and 8 on the inner circle. (This allows you to cut 8 portions easily without cutting through the tomatoes.)

Bake until the filling is firm to the touch and lightly colored, about 35 minutes. Cool in the pan on a rack for 20 minutes, then remove the ring and slide the tart onto a serving plate. Serve warm or at room temperature.