Heat 1 tablespoon of oil in large skillet over medium heat. Add onion and sauté until tender, about 5 minutes.
Add mushrooms and continue to cook, stirring frequently, until mushrooms are tender and onions are translucent. Remove from heat and set aside.
Unroll phyllo pastry and cover with plastic wrap or a damp towel to keep it from drying out while you work.
Place one sheet of phyllo on a large cutting board and brush lightly with oil. Layer another sheet of phyllo directly on top of the first one, and coat with another light layer of oil. Repeat this process until you have six layers of phyllo on top of each other.
Lightly grease a muffin tin.
Gently cut phyllo into six squares, making them roughly an inch larger than the size of your muffin cups, so the phyllo can fill each cup and run all the way up the sides.
Cut ham slices into circles small enough to fit in the bottom of each muffin cup.
Place phyllo gently into each cup, then add a ham slice to the bottom of each one.
Add spinach, feta, and sautéed mushrooms and onions to each cup.