Place butter in a mixing bowl, and using an electric hand mixer beat until light and fluffy. Scrape down sides of the bowl, add 1/4 cup sugar and continue to beat for 1 minute. Add zest from 1 meyer lemon and vanilla and mix together until just combined. Scrape down sides of the bowl once more.
In anther mixing bowl whisk together flour, tapioca starch and salt. Add flour mixture to the butter mixture and beat until dough forms.
Evenly press mixture onto a sheet of wax paper and form a 12″ x 4″ (1/4″ thick) square. Wrap the cookie dough slab completely in wax paper and refrigerate for at least 1 hour and up to 2 days.
Preheat oven to 350˚F.
Remove from fridge and smooth the surface of the dough with a rolling pin if needed (easier to do this once the dough is cold and stiff). Unwrap the dough and cut cookies into 1″ x 1.5″ rectangles (if dough gets too soft to handle while cutting them up and transferring to a baking sheet, refrigerate for 30 minutes before continuing). Using a small offset spatula, place cookie dough slices onto a parchment lined baking sheet and bake cookies for 12 to 15 minutes or until the edges barely begin to brown. Remove from oven and allow cookies to cool, 15 to 20 minutes.
While cookies cool, make the icing. Place powdered sugar in a mixing bowl and gently stir in lemon juice and milk until completely combined and no lumps remain.
Loosely drizzle icing over cooled cookies and top with a couple strips of zest. Allow icing to set, 30 minutes to 1 hour and serve.