Combine 1 1/2 cup flour, 2 tablespoons sugar, and salt in a large bowl. Cut in butter using 2 knives or a pastry cutter until mixture resembles coarse crumbs.
Mix water and vanilla extract together in a small bowl. Stir into flour mixture, 1 tablespoon at a time, until dough comes together. Gently roll into a ball and flatten slightly. Wrap with plastic wrap and chill until firm, about 30 minutes.
Toss apples with 1/2 cup sugar and 3 tablespoons flour in a large bowl.
Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Roll dough out to a 12-inch circle on a floured work surface. Transfer to the baking sheet. Arrange apple slices in the center of the circle, leaving a 2-inch border all around. Fold up border over apples, pinching together slightly.
Combine 1 tablespoon sugar and cinnamon in a small bowl. Sprinkle over apples.
Bake crostata in the preheated oven until crust is golden brown, 30 to 40 minutes.