Preheat oven to 325 degrees. Line cookie sheet with baking parchment or grease lightly.
Combine flour and cream of tartar.
Cream shortening with sugar and salt until fluffy. Add egg, milk, baking soda and vanilla. Mix well. Add flour and mix until well blended. Pat dough into a large ball.
Dust hands with flour. Pinch off golf-ball-size pieces of dough. Place on cookie sheet and flatten into slabs 3/8-inch thick. Brush with more milk if a glossy top is desired. Bake until brown and almost firm, about 25 minutes. Check to be sure cookies don’t burn. Cookies will harden as they cool. Makes about 2 dozen.