Beat butter and cream cheese together for 1-2 minutes. Add sugar and beat for another 2-3 minutes. Add egg and vanilla and beat for another 3-4 minutes.
Add the lemon zest and lemon juice and mix for 1 minute.
In a separate bowl, combine the flour, cornstarch, baking soda and salt.
Mix the flour mixture into the wet ingredients. The dough should start to form a ball, so if the dough seems too wet, you can add another 1/4 cup and keep mixing.
Refrigerate the dough for at least 3 hours or up to 3 days.
Preheat oven to 350° and line a cookie sheet with parchment paper or a silicone mat. Use a cookie scoop to scoop out heaping tablespoonfuls of dough. Place on a prepared cookie sheet about 2 inches apart.
Bake for 8-10 minutes. If it looks like the cookies are not quite done, you may need to bake them another 2-3 minutes, depending on your oven and the size of your cookies.
Let cool on cookie sheets for about 5 minutes before transferring to a wire rack to cool completely. Once the cookies are cooled, make the glaze.
Mix glaze ingredients together, adding a little more lemon juice if needed for the desired consistency. You can drizzle the frosting over the top or you can dip the tops of the cookies in the glaze. Allow it to harden and then store the cookies in an airtight container.