Preheat the oven to 350°F. Cream together the butter, oil, sugar, powdered sugar, water, eggs, and vanilla until smooth and well combined. In a large bowl, whisk together the flour, salt, baking soda and cream of tartar. Add the dry ingredients to the butter mixture 1 cup at a time, mixing after each addition, until all of the dry ingredients have been incorporated into the dough.
Roll the dough into golf ball-size balls and place them 2 inches apart on a cookie sheet. Dip the bottom of a glass (about 2 1/2 inches wide) in some granulated sugar and then press down on one of the cookie dough balls until it is 1/2” thick. This will leave a flat middle section with the signature rough edge. Repeat with the rest of the cookie dough balls.
Bake the cookies for 8-10 minutes, or until the bottom is lightly browned. They stay soft, so make sure not to over-bake them. Let them cool for 5-10 minutes on the cookie sheet then move them to a wire rack to cool completely.
To make the frosting, cream together the butter, sour cream, and vanilla until well combined. Add the powdered sugar, one cup at a time, mixing well after each addition. Add 2-3 tablespoons of milk to thin the frosting to your desired consistency. Add the food coloring, if using, and whip the frosting for one minute. Spread about 1 tablespoon of frosting on each cookie, allow to set, and then store the cookies in an airtight container in the refrigerator.