Sift together the flour, baking powder, sugar, cream of tartar, and salt; set aside. Combine the eggs, milk, and vanilla; set aside.
In a large mixing bowl, beat shortening for 30 seconds on medium to high speed. Alternately add dry ingredients and egg mixture to beaten shortening; beat well after each addition. Divide dough in half. Cover and chill for 3 hours or until easy to handle.
Preheat the oven to 350 degrees F. On a lightly floured surface, roll dough, half at a time, to 1/4-inch thickness.
Using 4- to 5-inch cookie cutters, cut the rolled dough into desired shapes. (Between cuts, dip cutters into flour to keep dough from sticking to them.) Place cutouts on ungreased cookie sheets. For ornaments, before baking, use a drinking straw to cut a hole near the top of each cutout. (After baking and decorating, thread twine through the holes.) For lollipops, leave enough room for lollipop dowels on baking sheets. Insert dowels into bottoms of cookies, pushing each dowel 1 1/2 inches into dough.
Bake in the preheated oven for 12 to 14 minutes or until the edges are firm and the bottoms are lightly browned. Cool on a wire rack. Spread a base of tinted or white Royal Icing over the cookies and allow to set. Decorate as desired. Makes 15 large cookies.
Royal Icing: In a bowl, beat together 3 3/4 cups sifted powdered sugar, 3 tablespoons meringue powder, and 6 tablespoons warm water for 3 to 4 minutes on high speed. Divide into bowls and add food coloring to tint as desired. To spread cookies with a base coat of icing, add water, 1/2 teaspoon at a time, to the base icing to make of spreading consistency. For piped decorations, use a decorating bag fitted with a No. 2 tip. Keep bowls of icing tightly covered with plastic wrap when not in use.