Add eggs to a saucepan and cover with cold water. Bring to a boil over high heat. Turn off heat and let sit covered for 10 minutes. Drain, cool, peel and chop.
Heat oil in a large skillet over medium. Add 1 tortilla and top with half of the eggs, cheese, ham, mushrooms and parsley. Top with another tortilla and cook until golden on both sides, flipping once. Repeat with remaining tortillas and toppings. Quarter the quesadillas.
Serve each family member 2 quesadilla quarters with sliced plums.