Place butter in a large bowl and cream using an electric mixer for 3 minutes. Add confectioners' sugar and mix until incorporated. Add each egg individually to the creamed butter, scraping the sides of the bowl if needed. Add vanilla extract and mix for 3 more minutes.
Sift flour, baking soda, and cream of tartar together in a bowl. Add flour mixture to the butter mixture, mixing on low speed. Cover and chill the dough until it is workable, 2 to 4 hours.
Preheat the oven to 350 degrees F (175 degrees C).
Roll dough out onto a work surface and cut using small animal-shaped cookie cutters. Place cookies on a baking sheet.
Bake in the preheated oven on the middle rack until crisp, 7 to 9 minutes. Let cool completely before icing, about 45 minutes.
Mix 2 cups confectioners' sugar, half-and-half, and vanilla extract together in a bowl. Divide icing and add pink food coloring to 1 half. Dip cookies in icing and decorate with sprinkles quickly before icing dries. Place cookies on a rack to let icing dry to a shiny finish.
