Preheat oven to 350°F. Line a sheet pan with parchment paper or a silicone baking mat.
In the bowl of a stand mixer fitted with a paddle attachment, or in a bowl with a hand-held mixer, beat the butter and sugars together until smooth and creamy, about 3-5 minutes. Add the egg and vanilla extract and mix until just combined.
In a separate bowl, combine the flour, baking soda, salt, and cream of tartar. Add to the mixer and mix on low until combined. Mixture will be crumbly at first, but will eventually come together into a thick dough.
Portion into rounded tablespoons, roll into a ball, and place on the prepared sheet pan 2 inches apart. Use your hand or the bottom of glass to press the balls into thick disks.
Bake until bottoms just turn golden brown, about 11-13 minutes (see Notes). Let cool on the pan 5 minutes, then transfer to a sheet pan to cool completely.
To make the frosting:
Beat the butter, powdered sugar, vanilla extract, almond extract, pinch of salt and 1 tablespoon of the milk/cream until smooth and creamy. Add more milk 1 tablespoon at a time until spreadable consistency is reached, pausing to scrape down the bowl as needed.
Add food coloring, if using. Frost cookies and decorate as desired.
Store cookies in an airtight container at room temperature for up to 1 week. Cookies also freeze very well for up to a month or more. Thaw before serving.
Video
Notes
To roll and cut cookies into shapes: Do not chill. Roll out on a sheet of parchment paper to ¼-inch thick and cut out shapes as desired. Place on sheet pan lined with parchment paper and bake (if you're worried about spreading, freeze cut out cookies on sheet pan for 15 minutes). Baking time will vary based on cookie size.
To make Almond Sugar Cookies: Substitute 1 teaspoon almond extract for the vanilla extract in the cookies. Omit vanilla in frosting and increase almond extract to ½ teaspoon.
If you wait for the edges of the cookies turn brown, they are already a little overdone. What I do when I try a new cookie recipe is I bake one test cookie, checking it often. And then I bake the rest when I determine the correct baking time. These cookies turned out with a barely golden bottom at exactly 12 minutes in my oven.