Add the first 9 ingredients to a 2-pound bread machine according to the manufacturers directions. Select the dough cycle. When cycle is complete, remove dough from machine. Punch down. Cover and let rest for 10 minutes.
Meanwhile, for filling, in a food processor bowl or blender container combine berries; cover and process or blend until nearly smooth. Press the berries through a sieve and discard seeds. In a medium saucepan stir together the 1/3 cup sugar and cornstarch. Stir in sieved berries. Cook and stir until thickened and bubbly. Stir in lemon peel.
Divide the dough in half. Press half of the dough into a greased 12-inch pizza pan, stopping and letting dough rest a few minutes occasionally. Spread with the filling to within 1/2 inch of the edges. On a lightly floured surface, roll the remaining dough into a 12-inch circle; cut into 3/4-inch-wide strips. Weave the strips over filling in a lattice pattern. Trim ends as necessary; press against bottom dough to seal.
Cover and let rise in a warm place about 30 minutes or until nearly double. Bake in a 375 degree F. oven for 25 minutes. If necessary, loosely cover with foil last 10 minutes to prevent overbrowning. Cool slightly in pan on a wire rack. Dust with powdered sugar. Makes 16 servings.