Combine butter, sugars, eggs, flavoring, and salt. Beat until just combined. Add flours, 1 cup at a time. If dough is sticky to the touch,add more self-rising flour in 1/4 cup increments until smooth. Stir in walnuts.
Scoop onto a baking sheet one dozen at a time. Bake in center of oven 10 to 13 minutes, until cookies are puffy and beginning to turn tan. Remove from oven and let cool.
Meanwhile, mix icing ingredients until texture is smooth and thick. Add more milk or sugar to achieve a creamy texture. Ice cookies. Makes 2 dozen.