Preheat oven to 350F/176.7C. Line a large cookie sheet or baking sheets (17" x 12") with parchment paper. Or use two baking sheets.
MAKE COOKIE DOUGH: In a food processor, process butter and icing sugar until creamy smooth, about 1 minute. Add vanilla, cornstarch, lemon zest and flour. Pulse with quick on/off turns just until dough starts to get a crumbly texture and ingredients are combined. Empty onto a cutting board or parchment paper and gently shape shortbread dough together into one big blob. If you don't have a food processor, use an electric mixer on low speed or a stand mixer with a paddle attachment.
1 cup room temperature salted butter (2 sticks) , 1/2 cup icing sugar, 1/2 cup corn starch, 1 1/2 cup all purpose flour, Note 2, 1/2 teaspoon pure vanilla extract, 1 teaspoon lemon zest, Note 2
FORM COOKIES: Using clean hands, break off a small piece of dough, roll into a 1 1/2 inch/3.8 cm ball. Place on prepared baking sheet. Repeat until all dough is gone. I make 24-30 balls (24 will fit in one large pan). With tines of a fork, flatten balls to just over 1/4 inch/6.3mm and make a criss-cross pattern. (Tip: dip fork into flour if dough is sticky).
BAKE AND COOL: Bake cookies in middle rack in preheated oven for 16 minutes or until edges are starting to turn golden brown. Transfer to cooling rack.
MAKE GLAZE: While cookies are baking, mix glaze ingredients together with whisk or fork. When cookies are cooled (if you can wait!), brush glaze on each cookie with a pastry brush. Do this over parchment paper to catch the drips. Sprinkle on extra lemon zest if you like. Glaze will harden in about 30 minutes.