Preheat oven to 350 degrees F.
In a medium-sized mixing bowl, mix together flour, salt, baking soda, and cream of tarter; set aside.
In the bowl of a standing mixer, cream butter and both granulated & powdered sugars together for 2 minutes until fluffy. Slowly stream in the oil while beating. Add the lemon juice, lemon zest, and vanilla and then beat in the eggs one at a time until well combined.
Slowly add the flour mixture to the wet ingredients, mixing until combined (scrape down the sides of the bowl as needed).
Roll dough into 2 inch balls to make approx. 36 cookies. Place ¼ cup sugar in a small dish. Place rolled dough on a baking sheet about 3 inches apart. Using a drinking glass (or something that is 2.5 inches in diameter), dip into sugar. Press the glass into a ball of dough until it is ½ inch thick (the dough will form a jagged edge and spill past the edge of the glass). Repeat the process of dipping the glass into sugar and pressing it into a ball of dough.
Bake for 8 minutes only. Gently remove cookies from pan onto a cooling rack and let them cool completely. Place them in an air tight Tupperware in the fridge.
Cream butter and sour cream.
Slowly add powdered sugar a little at a time, alternating with adding the lemon juice. Add extra lime juice or powdered sugar if needed to reach desired consistency.
Lastly, add food coloring and vanilla; mix until combined.
Spread over cold cookies and serve immediately or let the icing dry (it will dry to the touch) and then store in an air-tight container in the fridge.
Adapted from Garnish & Glaze