In a large bowl, beat butter until creamy. Add powdered sugar and mix until light and fluffy. Add lemon zest and vanilla extract, beat well.
Add flour and cornstarch and mix well until thoroughly combined.
Using your hands, roll cookie dough into 1-inch balls. Place on an ungreased cookie sheet and bake for 15 minutes, until bottoms are light brown, remove and cool on wire cooling racks.
Once cool, spread lemon frosting onto cookies.
In a medium bowl combine butter, lemon zest, lemon juice and powdered sugar. Stir until well mixed. Additional lemon juice may be needed to get the frosting thin enough, then dip the cookie into the frosting.