Preheat oven to 350 degrees. Beat brown sugar and margarine at medium speed of an electric mixer until light and fluffy. Add egg, milk, and vanilla; beat well. Combine 3 cups flour, baking powder, and salt, stirring well. Gradually add flour mixture to creamed mixture, mixing well. Cover and chill at least 2 hours. Divide dough in half. Work with half at a time, storing remainder in refrigerator.
Sprinkle 1 tsp. flour evenly on work surface. Turn dough out onto floured surface, and roll dough to 1/8 inch thickness. Cut with a 2 inch cookie cutter and place 2 inches apart on cookie sheets coated with cooking spray. Bake at 350 degrees for 6-8 minutes or until edges of cookies are lightly browned. Remove from cookie sheets, and let cool on wire racks. Repeat procedure with remaining flour and remaining half of dough. Combine powdered sugar and water. Pipe frosting around the edges of half the cookies; spread frosting on the tops of remaining cookies.