Preheat the oven to 350 degrees Fahrenheit. Line a baking sheet with parchment paper or a silicone liner.
In a stand mixer bowl (or other large bowl), cream together your butter, shortening, granulated sugar, and powdered sugar for about 2 minutes till light and fluffy.
Add in the milk, eggs, and vanilla extract, and stir until combined.
Add in the flour, baking soda, cream of tartar and salt and stir till it all comes together. The dough should come together, but may be a little crumbly.
To make large cookies: Scoop the dough into heaping 3 Tablespoon sized scoops, and roll into balls.To make small/more normal sized cookies: Scoop the dough into heaping 1.5 Tablespoon sized scoops, and then roll into balls. Place cookies onto the prepared baking sheet. (Only about 6 cookies for the large cookies, and 12 for the small).
Add the 1/4 cup of granulated sugar for pressing the cookies onto a small flat plate.
Spray the bottom of a flat circular glass with nonstick spray. Press it into the sugar.
Press each cookie down using the glass cup, rocking the cup a bit side to side, until they're about 1/2 inch thick. Dip the cup back into the sugar in between each cookie.
Bake the cookies in the preheated oven until the cookies look set on the sides, and barely done on top. They may not look cooked through the middle, but they will keep cooking on the tray. Cook large cookies for 8-10 minutes. Cook the small cookies for 6-8 minutes.
Allow the cookies to cool for several minutes on the baking pan. Then transfer them to a cooling rack to cool completely.