Preheat oven to 375 degrees F. In a large bowl, combine butter and shortening. Beat with an electric mixer on medium to high speed for 30 seconds. Add the 1 cup sugar, the coffee liqueur, baking soda, cream of tartar, and salt. Beat until combined, scraping side of bowl occasionally. Beat in egg and vanilla until combined. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour.

In a small bowl, combine the 1/4 cup sugar and the 2 teaspoons cinnamon. Shape dough into 1-inch balls. Roll balls in cinnamon-sugar to coat. Place balls 2 inches apart on an ungreased cookie sheet.

Bake in the preheated oven for 8 to 10 minutes or until edges are lightly browned. Transfer cookies to a wire rack; let cool.

Spread Irish Creme Frosting over cookies. If desired, sprinkle with additional cinnamon.

Irish Creme Frosting: In a medium bowl, combine 3 tablespoons softened butter and 1 tablespoon coffee liqueur. Beat with an electric mixer on medium to high speed until combined. Gradually beat in 3 cups powdered sugar. Beat in enough half-and-half or whipping cream (1 to 3 tablespoons) to make a frosting of spreading consistency.

To Store: Layer cookies between waxed paper in an airtight container; cover. Store at room temperature for up to 3 days or freeze for up to 3 months.