In the bowl or jar, mix together the oil, garlic, oregano, thyme, and basil. Set aside.
Warm the skillet over medium heat and add 1 tablespoon of the herb oil to the pan. Saute the onions for a couple minutes, until they soften and glisten. Add the spinach and mushrooms, sauteeing just long enough for the spinach to wilt. Remove from heat and drain vegetables as necessary. Season with salt and pepper.
Shape the dough into two pizzas. Brush the grill with oil and place the shaped doughs on the grill. Cook for about 3 minutes, until grill marks show and the bottom is crusty. Pick up the crusts with the tongs and place on a peel or cookie sheets, cooked side down.
Brush on some of the herb oil, and sprinkle on the grated Parmesan. Add the cooked spinach, onions, and mushrooms. Drop small dollops of the pesto around the pizza and sprinkle on the crumbled feta.
Carefully transfer the pizza back to the grill and cook for 5 to 8 minutes, until the bottom is crusty and browned and the toppings are hot. Use the tongs to move the pizza around the grill for even cooking. Remove from grill, slice, and serve.