Cream butter, sugar, and lemon zest until pale and fluffy, 2-3 minutes.
Beat in egg, lemon juice, and vanilla extract.
Whisk flour, cornstarch, and salt; mix into wet ingredients until just combined.
Chill dough 20-30 minutes.
Roll into 1-tablespoon balls and press a deep well into each.
Chill shaped cookies 15 minutes more.
Preheat oven to 350°F. Bake 11-13 minutes, until edges are just barely golden.
Cool completely, then fill each well with 1/2 tsp lemon curd.
Whisk powdered sugar and water into a thin icing; drizzle over cookies and let set 10 minutes.