For rectangular cookies, lightly butter the bottom of a straight-sided 9x13-inch baking pan. For wedges, lightly butter two 9-1/2-inch fluted tart pans with removable bottoms.
Combine the flour, cornstarch, and salt in a medium bowl. In a stand mixer fitted with the paddle attachment, beat the butter, sugar, and vanilla on medium speed until smooth and creamy, 2 to 3 minutes. Add the flour mixture and mix on low speed until the dough begins to form moist clumps, about 1 minute.
Turn the dough out into the prepared pan (if making wedges, divide the dough between the two pans). Using lightly floured fingertips, press the dough into the pan(s) to form an even layer.
For rectangular cookies, use a bench knife or a paring knife to cut the dough all the way through into 40 bars. With the tines of a fork, prick each bar twice all the way through, spacing the holes evenly and on the diagonal. Lightly flour the tines as necessary to prevent the dough from sticking.
For wedges, cut each pan of dough all the way through to form 12 wedges. With the tines of a fork, prick the wedges three times all the way through, spacing the holes evenly over the dough.
Freeze the dough while the oven heats.
Position a rack in the center of the oven and heat the oven to 325°F.
Bake the dough, rotating the pan(s) halfway through, until the top(s) looks dry and golden-brown, 30 to 40 minutes. Transfer the pan(s) to a rack. Using a paring knife or bench knife, immediately re-cut the cookies along the lines. Cool completely before removing them from the pan(s).
Once the dough is pressed into the pan and cut, it can be wrapped and refrigerated for 1 day or frozen for up to 1 month. If freezing the dough, thaw it overnight in the refrigerator before baking.
Baked and cooled cookies can be layered between sheets of parchment or waxed paper in an airtight container and stored at room temperature for up to 4 days.
For chocolate-dipped shortbread: Melt 6 oz. chopped bittersweet or white chocolate with 1 tsp. canola or vegetable oil. Brush any lose crumbs from cooled shortbread and dip two-thirds of each cookie into the chocolate. Dry on a cooling rack until chocolate firms, about 45 minutes.