Heat oven to 375°F. Slice 1 vanilla bean lengthwise with sharp knife on flat surface. Scrape seeds into large bowl; reserve bean pod shell. Add 1 cup butter to seeds in bowl; beat at medium speed until creamy. Add sugar, 1/2 cup powdered sugar and egg; continue beating until well mixed. Reduce speed to low; add flour, baking soda, cream of tartar and salt. Continue beating until well mixed. Stir in walnuts.
Place additional sugar into small bowl. Shape dough into 1-inch balls; roll balls in sugar. Place 2 inches apart onto ungreased cookie sheets. Bake for 9 to 11 minutes or until cookies are lightly browned. Cool completely.
Meanwhile, slice 1 vanilla bean for frosting lengthwise with sharp knife on flat surface. Open vanilla bean to expose seeds; place vanilla bean with seeds and reserved vanilla bean pod shell into 2-quart saucepan. Add half & half; cook over medium heat until mixture just starts to boil. Remove from heat; let mixture stand at room temperature until cool (30 minutes). Discard vanilla bean pods.
Combine half & half mixture, 2 tablespoons butter and 2 cups powdered sugar in small bowl. Beat at low speed, adding additional half & half if necessary for desired frosting consistency. Frost cooled cookies.
*Substitute 1 tablespoon real vanilla extract.
**Substitute 1 tablespoon real vanilla extract. Eliminate step to cook vanilla bean in half & half. Combine half & half with butter and powdered sugar as directed.
To cut open a vanilla bean, lay it flat on a cutting surface. Holding one end of the bean to the surface, carefully slice the bean open lengthwise. When separated, thousands of tiny seeds are exposed. This is why it is technically a seed-pod rather than a bean. By cutting the bean open before placing it into liquid, more of the surface of the bean is exposed providing more vanilla flavor.
Regardless of where the vanilla beans come from, premium beans should have a full rich aroma, be oily to the touch, and sleek in appearance. Beans to avoid are those with very little scent, are smoky, brittle or dry or are have mildew.
Vanilla beans will keep indefinitely in a cool, dark place in container with tight-fitting lid. Don’t refrigerate the beans as this can cause them to crystallize.