In a large bowl, cream butter until light and fluffy. Add sugar. Beat in egg and vanilla. Combine the flour, baking powder, and salt; add to creamed mixture and mix well.
Form dough into 1/2-1″ balls and placed on cookie sheets. Flatten with the bottom of a glass dipped in flour. Bake at 375 degrees for 6-8 minutes or until set but not brown. Cool on wire racks.
For frosting, cream butter. Add powdered sugar and vanilla; it will be very stiff. Add 3-4 tablespoons of milk, 1 tablespoon at a time until spreading consistency is reached. Add food coloring to a portion or all of frosting if desired. Frost cookies.