In large bowl, beat butter on medium speed until smooth and creamy (about 1 minute). Beat in powdered sugar and vanilla. Blend in 2 cups flour until a soft, crumbly dough forms. Press mixture evenly onto bottom of a lightly greased 9x13-inch baking pan. Bake in a preheated 350 F oven until light golden brown (18-20 minutes).
In small bowl, combine eggs, sugar, lemon juice, ¼ cup flour, lemon peel and baking powder; beat until well blended (about 2 minutes).
Sprinkle berries over warm crust. Pour filling over evenly arranged berries with a spoon. Bake until filling is lightly browned and set (30-35 minutes). Cool completely. Cover and refrigerate 4 hours or overnight.
To Serve: Sprinkle with powdered sugar. Cut into bars. Place in covered shallow container for transporting.