Heat a large nonstick skillet over medium-high heat. Coat pan with cooking spray. Add onions and shallots; cook 4 minutes or until golden. Remove onion mixture from pan; set aside. Add spinach to pan; cook 6 minutes or until wilted, stirring occasionally. Place spinach in a colander. Add arugula and beet greens to pan; cook 3 minutes or until wilted. Place arugula mixture on top of spinach in colander; drain, pressing until barely moist.
Combine onion mixture, spinach mixture, and cheeses in a large bowl. Stir in parsley, dill, salt, fennel seeds, and pepper.
Place one sheet of phyllo dough on a large cutting board or work surface (cover remaining dough to keep from drying); lightly coat with cooking spray. Place sheet in a 10-inch deep-dish pie plate coated with cooking spray, gently pressing into bottom and up sides of pan, letting ends of phyllo overlap the top of pan. Repeat procedure with remaining phyllo sheets, placing in a crisscross pattern. Spoon spinach mixture evenly over phyllo. Fold overlapping ends of phyllo toward center of pan. Coat ends with cooking spray, and gently press to hold shape.
Bake at 375° for 40 minutes or until golden. Cool 15 minutes on a wire rack; cut into wedges.