Preheat oven to 350 degrees and very lightly grease a baking sheet.
In a large bowl, cream together butter, vegetable oil, 1 1/4 cups sugar, powdered sugar, milk, and eggs.
In a second large bowl, whisk together flour, baking soda, cream of tartar, and salt.
Add dry ingredients to wet ingredients and mix until dough comes together.
Roll dough into balls (roughly the size of golf balls) and place on greased baking sheet 2-3 inches apart.
Place remaining 1/4 cup sugar in a shallow dish. Press the bottom of a tall drinking glass into the cookie dough in the bowl (just to moisten the bottom of the glass, no cookie dough should stick to it), then dip into sugar. Press the bottom of the glass into the top of each cookie dough ball to flatten about halfway. (The edge should "break" to create the signature crinkle edge.)
Bake for 8-11 minutes until cookies look dry but not browned. Allow to cool on baking sheet for 4-5 minutes, then transfer to a cooling rack to cool completely. Repeat process with remaining cookie dough.
To make the frosting, melt butter in a medium sauce pan. When butter is melted, continue to whisk over medium heat until butter turns from yellow to a golden color. Remove from heat. In a medium bowl cream together butter, cream cheese, powdered sugar, vanilla, and milk until smooth and spreadable.
Divide the frosting between three bowls. Whisk yellow food coloring into one bowl, and red and yellow into a second bowl to make orange frosting. Spread white frosting on 1/3 of one cookie, followed by orange and yellow frosting to cover the cookie. Repeat with remaining cookies, serve with an 8-ounce glass of milk.