In a large skillet over medium-high heat, heat the butter and oil until the butter is melted. Add the mushrooms and garlic and saute for 2 minutes. Pour in the Marsala and season with salt and pepper. Continue cooking until the mushrooms are cooked down and the liquid is mostly gone, 5 to 7 minutes. Mix in 2 tablespoons of the parsley. Set aside.
Put the flatbreads on a baking sheet. Brush with the pesto, then top with the mozzarella, mushroom mixture and goat cheese. Sprinkle with the remaining 1 tablespoon parsley.
Bake until the flatbreads are heated through and the cheese is melted, bubbly and beginning to brown, about 15 minutes.