Preheat oven to 375°F.
Heat oil in a nonstick skillet over medium heat. Add onions and sauté until translucent, three to five minutes. Add garlic. Cook another two minutes.
Transfer to a large bowl. Make a small hole in the bag of spinach and squeeze the bag to remove as much liquid as possible. Add spinach, feta cheese, mozzarella cheese, salt, dill, pepper, eggs, and egg whites to the bowl.
Spray a 9- or 10-inch springform pan with nonstick cooking spray. Place the mixture in the springform pan.
Cut the defrosted phyllo sheets into thin strips, like ribbons, some short and some longer. Place them over the spinach mixture in many different directions so that it looks pretty.
Spray the top with nonstick cooking spray (so that when it bakes the phyllo dough will acquire a golden color). Bake for 35 minutes.