Preheat oven to 350°.
Place walnuts in your food processor and pulse for about 20 seconds; set aside.
In a large bowl, combine flour, baking soda and salt and set aside.
In a separate bowl, cream butter and sugar together, then add egg and vanilla and stir to combine. Next, combine wet and dry ingredients. Be patient with this step -- this could take 30-60 seconds until the dough comes together. If the dough is still too dry, add 1 tablespoon milk.
Stir HALF of the crushed walnuts (1/2 cup) into the dough. Roll dough into balls then roll the balls into the remaining crushed walnuts and place them 2 inches apart on a lined baking sheet.
Bake cookies for 10 minutes; then, remove them from the oven, wait 1-2 minutes for them to slightly cool, and carefully use your thumb to press a thumbprint into the center of each cookie. Place cookies back into the oven and bake for an additional 3-5 minutes. Remove cookies from oven and onto a cooling rack.
While cookies are cooling, make the buttercream. Whisk butter and milk together, then add the powdered sugar and stir until smooth. If you choose to add food coloring, do so as this time. Wait until the cookies have cooled before adding the buttercream.
You may pipe the icing or simply use a spoon to scoop it into the center of each cookie.