Set a pizza stone on an oven rack and heat oven to 500° for about 45 minutes (if you don't have a pizza stone, set a large baking sheet on a rack and heat until oven is hot).
Put 2 1/4 cups flour, kosher salt, and oil in a food processor. With motor running, slowly add 1/2 cup water. Dough will come together into a ball and should feel soft and supple; if it is sticky at all, add more flour, 1 tbsp. at a time.
Divide dough into 12 portions. Using a floured rolling pin, roll 1 portion at a time on a well-floured work surface into a round about 8 in. wide and just thin enough to see through. Lightly sprinkle with sea salt and press it in with your hands. Prick dough all over with a fork (this will prevent the dough from puffing up too much).
Flour a wooden peel or back of a baking sheet generously and transfer dough to it. Gently slide dough onto hot pizza stone. Bake until matzo is light golden and crisp on each side and a bit darker at the edges, turning once with a wide spatula, 2 to 3 minutes total. Transfer matzo to a cooling rack and make remaining matzos the same way.
Rebake any matzo that isn't crisp in the center, which may be the case if they baked on a baking sheet; put matzos on a rimmed baking sheet, reduce oven heat to 250°, and bake 15 to 25 minutes more.