In a mixing bowl, mix together flour, salt, baking soda, and cream of tarter. Set aside.
In the bowl of a standing mixer, cream butter and sugars together for 2 minutes. Slowly stream in the oil while beating. Add the water and vanilla and then beat in the eggs one at a time until combined. Add the flour mixture to the wet ingredients a little at a time, mixing until combined (scrape down the sides of the bowl as needed).
Preheat oven to 350 degrees F.
Roll dough into 2 inch balls. I used an ice cream scoop to portion the dough. You should get 26 balls of dough.
Place 1/4 cup sugar in a small dish with a pinch of salt.
Place balls of dough on a baking sheet about 3 inches apart (I fit 8 on a baking sheet). Take a drinking glass or something that is 2.5 inches in diameter and dip it into sugar (you may need to press it into some dough first in order for the sugar to stick to the glass). Press the glass into a ball of dough until it is 1/2 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. You don't want them to be too done. Gently remove cookies from pan onto a cooling rack and let them cool completely. Place them in an air tight Tupperware in the fridge.
For the Icing:
Cream butter, sour cream, and salt together. Add powdered sugar a little at a time, alternating with adding the milk. Add extra milk or powdered sugar if needed to reach desired consistency. Add food coloring and vanilla and 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. I like to put wax paper between when stacking them.