In a stove top pan, heat the olive oil and add the mushrooms and onion. Season with salt and pepper to taste and cook over medium-high heat for around 4 minutes then lower the heat to medium. Continue to cook for another 6-8 minutes. Add balsamic and pepper flakes. Cook 2 minutes longer. Remove from heat and cool slightly.
To assemble: Preheat oven to 425°F.
Press the pizza dough onto a baking sheet lined with parchment paper into roughly an 13 x 9 inch rectangle.
Spread the crust with pesto leaving about ½ inch of crust around the edge.
Continue to cook for another 15 minutes until the onions begin to caramelize and the mushrooms have browned.
To assemble, Sprinkle ½ of the mozzarella cheese over the pesto, then layer the mushrooms and onions. Sprinkle with the feta, the remaining mozzarella and Parmesan cheese.
Place into the oven and bake for 15-17 minutes.
Garnish with parsley, then use a pizza cutter to slice into 8 full size or 32 appetizer size pieces.