Bring 2 cups water to a boil in a Dutch oven. Add kale; cover and cook 4 minutes. Stir in Swiss chard; cover and cook 3 minutes. Drain well, and cool. Place greens on paper towels, and squeeze until barely moist. Place greens in a large bowl; set aside.
Heat 2 teaspoons olive oil in pan over medium heat. Add onion and fennel; sauté 10 minutes or until soft. Stir onion mixture, rice, and next 8 ingredients (rice through feta) into greens. Set aside.
Preheat oven to 350°.
Gently press 1 sheet of phyllo into an 11-inch round removable-bottom tart pan coated with cooking spray, allowing ends to extend over edges of pan; lightly coat with cooking spray. Place second sheet of phyllo over first sheet to form a crisscross design; lightly coat phyllo with cooking spray. Repeat procedure with 5 sheets of phyllo and cooking spray, pressing sheets down into pan. Spoon greens mixture over phyllo in pan. Cover greens mixture with 1 sheet of phyllo, continuing crisscross design; lightly coat phyllo with cooking spray. Repeat procedure with 2 sheets of phyllo and cooking spray. Cover with remaining sheet of phyllo. Fold in edges of phyllo to fit pan and form a rim. Brush with 1 teaspoon oil. Cut 5 slits in top of dough to allow steam to escape. Bake at 350° for 50 minutes or until golden brown. Let stand for 5 minutes.
Note: If you can find packages of fresh phyllo dough, the delicate sheets will be easier to work with (they separate more readily) than the frozen variety. Look for fresh phyllo dough in ethnic markets or specialty-food shops.