Make the dough: In a medium bowl, whisk together the flour, cornstarch, and salt until combined. Set aside.
In a large bowl using a handheld or stand mixer with a paddle attachment, beat the butter and granulated sugar together on medium-high speed until creamed, about 1 minute. Add lemon zest, lemon juice, egg, and vanilla bean paste. Beat on high speed until combined, about 1 minute. Scrape down the sides and bottom of the bowl as needed.
Add dry ingredients to wet ingredients and mix on low speed until combined. Dough will be creamy. Cover and refrigerate for at least 3 hours, and up to 3 days.
Shape & coat the dough: Place granulated sugar in a small bowl. Scoop cold dough (about 1 tablespoon per ball), roll into balls, then coat in granulated sugar. Place balls 2 inches apart on lined baking sheets.
Fill the cookies: Use your thumb to make an indentation in the center of each cookie dough ball. Fill each with 1/2 teaspoon of lemon curd.
Preheat the oven to 350°F (177°C) while the cookies chill in the refrigerator. Bake cookies for 12–13 minutes, or until edges appear set and lightly browned. Let cool for 5 minutes, then transfer to a wire rack.
Make the icing: In a small bowl, whisk together confectioners’ sugar and lemon juice. Drizzle over cooled cookies and allow icing to set for 30 minutes.