Bring apricots and water to cover to a boil in a small saucepan. Reduce heat, and simmer, uncovered, 15 minutes or until tender. Drain and coarsely chop apricots; set aside.
Beat butter at medium speed with an electric mixer until creamy; gradually add 1/4 cup sugar, beating well. Stir in 1 cup flour, and press mixture into a lightly greased aluminum foil-lined 9-inch square pan.
Bake at 350° for 15 to 20 minutes or until lightly browned.
Beat eggs at medium speed until thick and pale; gradually add brown sugar, beating well. Add remaining 1/3 cup flour, baking powder, and salt, beating well. Stir in chopped apricot, walnuts, and vanilla; spread mixture evenly over crust.
Bake at 325° for 45 minutes. Let cool completely in pan. Cut into bars; sprinkle with powdered sugar.