Preheat oven to 375°.
Heat oil in a large nonstick skillet over medium-high heat. Add onions; sauté 4 minutes. Add fennel; sauté 3 minutes. Remove onion mixture from pan; cool. Add spinach to pan; sauté 30 seconds or until spinach wilts. Place spinach in a colander, pressing until barely moist. Add greens to pan; sauté 30 seconds or until greens wilt. Place greens in a colander, pressing until barely moist. Combine onion mixture, spinach, greens, feta, and next 5 ingredients (feta through black pepper) in a large bowl.
Working with 1 phyllo sheet at a time (cover the remaining dough to keep from drying), place 2 sheets in a 13 x 9-inch baking pan coated with cooking spray. Gently press sheets into bottom and sides of pan, allowing the ends to extend over edges of pan. Coat top sheet with cooking spray. Fold 1 sheet of phyllo in half crosswise; place on sheets in bottom of pan, and coat with cooking spray. Top with 1 sheet of phyllo, gently pressing sheet into the bottom and sides of pan; coat with cooking spray. Spread greens mixture evenly over top of phyllo. Fold a sheet of phyllo in half; gently press on the greens mixture in pan, and coat with cooking spray. Top with the remaining 3 sheets of phyllo, coating each with cooking spray. Cut ends extending over pan. Fold edges of phyllo to form a rim; flatten rim with fork. Cut 4 slits with a sharp knife in top of phyllo. Bake at 375° for 50 minutes. Cool for 30 minutes. Garnish with parsley sprigs, if desired.
Wine Note: This dish's green flavors deserve an herbal wine. Some of the world's most fresh, green wines are New Zealand Sauvignon Blancs; try Villa Maria Private Bin (about $12).