In a food processor, combine the flour, cornstarch, one-fourth teaspoon salt and butter. Pulse the mixture until it is in very tiny bits.
Add one egg and pulse until a dough forms. This dough is rather tough but, it does come together nicely.
Gather the dough into a ball. Then press the dough in to the tart pan, covering every bit. (You could, roll the dough out to a 12-inch circle, if you want.) Crimp the edges, and refrigerate for 30 minutes.
Roughly chop 1/4 cup of the basil leaves. Stack and roll the other basil leaves and cut them into thin ribbons.
Mix the cheese, mayo, chopped basil, green onions and pepper in a bowl. Spread the mixture over the bottom of the tart shell.
Top with grape tomatoes and press them down. Bake for 35 minutesï¿½until the crust edges are golden-brown and the cheese is bubbling up.
Cool for 10 minutes, then sprinkle with 1/4 cup sliced basil ribbons. Serve warm.