Preheat oven to 450◦. Lightly grease two baking sheets.
Place butter and sugar in the bowl of a stand mixer and beat until light and fluffy. Add the eggs and beat again. Add the milk and vanilla and beat until thoroughly mixed.
Add 1 1/2 cups of the flour along with the baking soda, cream of tartar, and salt to the mixing bowl and mix until combined. Add 2 cups more flour and mix again, then add remaining 2 cups flour and mix until fully incorporated, scraping down the sides as needed.
Divide dough in half. Roll one half out on a lightly floured surface to 1/8-inch thick. Cut out desired shapes with cookie cutters and transfer cookies to prepared baking sheet, leaving an inch or so between them. Re-roll dough scraps 1-2 times. Repeat with remaining dough half.
Bake one pan of cookies at a time just until the cookies puff up a little and set, about 4 minutes for 3-inch cookies. (You don't want the cookies to brown at all.) Let cookies cool on the pan for 4 minutes, then carefully transfer them to a flat surface until completely cool.