Spray a skillet with cooking spray; heat over medium. Toast both sides of each pita in skillet, cover, and set aside.
Remove skillet from stove and spray again with cooking spray. Reheat skillet over medium. Crack one egg into hot skillet and cook until whites are no longer clear and yolk begins to harden. Flip and cook for 1 minute longer (or more if firmer yolk is desired). Repeat with remaining eggs.
Mash avocado in a small bowl. Mix in fresh lemon juice, salt, and pepper. Spread mashed avocado over each pita, drizzle with olive oil, and top with an egg. Serve with oranges on the side.