Combine butter, chocolate chips, vanilla, and milk in top of double boiler.
Cook over low heat, stirring frequently, until all ingredients have melted together; remove from heat.
In large bowl, mix flour and walnuts; stir in chocolate mixture and refrigerate for 30 minutes.
Preheat oven to 350F and line two cookie sheets with foil.
Drop teaspoonfuls of dough onto prepared cookie sheets. Bake for 7 to 8 minutes or until cookies feel firm to the touch (they never look done). Let stand 2 minutes before removing from cookie sheets. Cool on wire racks.