Preheat your oven to 450 degrees with your pizza stone inside (if using).
While the oven is preheating, add the garlic and olive oil to a small bowl and allow to sit for at least 5 minutes (the longer the better).
Cut your morel mushrooms into small pieces, about the size of a dime. Sautee the mushrooms in butter over medium heat until just barely starting to crisp. Remove and let rest on a paper towel.
Carefully brush the entire flatbread with the garlic olive oil mixture, reserving about 1/2 a tbsp for the arugula.
Sprinkle the flatbread with the mozzarella cheese, then add the prosciutto. You can rip the prosciutto into smaller pieces to help distribute more evenly. Add your sauteed mushrooms, then sprinkle the Parmigiano Reggiano over top.
Transfer the flatbread to your preheated pizza stone or to a baking sheet. I keep it on the parchment paper to make transferring the flatbread a little easier. Bake for about 15 minutes, then check. You want the cheese to start bubbling and the outer crust to be golden brown. Remove the flatbread or bake another 3-5 minutes if needed.
While the flatbread is baking, pour the remaining garlic olive oil over the baby arugula and gently massage it into the leaves. Sprinkle with a pinch of salt.
Once you have removed the pizza from the oven, sprinkle the arugula over the top. Slice with a pizza cutter and serve.