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In a food processor process 3/4 cup of the toasted walnuts until very finely ground but still dry (not oily). Set ground walnuts and remaining finely chopped walnuts aside. In a large bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Add 1/2 cup of the powdered sugar. Beat until combined, scraping sides of bowl occasionally. Beat in milk, vanilla, and 1 teaspoon of the cinnamon until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour, the ground walnuts, and the finely chopped walnuts. Wrap dough in plastic wrap or waxed paper. Chill about 1 hour or until dough is firm enough to shape.
Preheat oven to 325°F. Shape dough into 2-inch balls. Place 2 1/2 inches apart on ungreased cookie sheets. Dip the bottom of a drinking glass with a distinctive design into granulated sugar; use glass bottom to flatten balls slightly.
Bake in the preheated oven about 15 minutes or until bottoms are light brown. Cool on cookie sheet for 2 minutes. Transfer to wire racks; cool completely.
Just before serving, in a small bowl combine the remaining 1/2 cup powdered sugar and remaining 1/2 teaspoon cinnamon. Sift powdered sugar-cinnamon mixture lightly over tops of cookies.