Preheat oven to 350 degrees. Line two baking sheets with parchment paper. Pour 1/4 cup granulated sugar into a small bowl for rolling cookies; set aside.
COOKIES: In the bowl of a stand mixer fitted with a whisk or paddle attachment, cream butter, oil, 1-1/4 cup sugar, 3/4 cup confectioners’ sugar, and water. Add eggs one at a time, beating until combined.
In a large bowl, whisk together flour, baking soda, cream of tartar and salt. Gradually add dry ingredients to the wet ingredients beating well after each addition.
Roll cookie dough into golf-ball sized balls. Roll dough balls in the sugar bowl and place 6 on each prepared baking sheet.
To give the cookies their signature rough edge, dip the bottom of a glass into the sugar bowl and gently press the glass into the center of the dough ball to flatten it a bit. You want the cookies to be thick, so don't press them too thin.
Bake 8-10 minutes or until bottom is lightly golden-brown. Do not over-bake. Set aside to cool. Repeat with remaining cookie dough.
FROSTING: Cream together butter, sour cream, vanilla and salt. Slowly add confectioners’ sugar and beat until desired consistency. Add milk to thin if needed. Beat in food coloring to desired shade of pink (a little goes a long way). Spread frosting over cooled cookies and serve!
To Store: Refrigerate frosted cookies on a baking sheet until the frosting has set. Once frosting has set, store cookies in covered containers. Stack in two layers with parchment or wax paper in between layers.
Cook’s Note: These cookies are traditionally served cold.