In a small bowl, cream butter and brown sugar. Beat in egg and sour cream; mix well. Combine the flour, baking soda and salt; gradually add to creamed mixture and mix well.
Drop by tablespoonfuls 2 in. apart onto greased baking sheets. Bake at 375° for 9-11 minutes or until golden brown. Remove to wire racks to cool.
For frosting, in a small saucepan, melt butter over low heat; add brown sugar. Cook and stir for 2 minutes. Gradually add the milk. Bring to a boil, stirring constantly. Remove from the heat. Stir in confectioners' sugar. Cool for 20-30 minutes. Frost cooled cookies.