For the pizza sauce: Add the tomato puree, olive oil, dried oregano, salt, pepper and 1 tablespoon water to a bowl and mix together.
For the pizza: Heat the olive oil in a large cast-iron skillet over high heat. Season the mushrooms with salt and pepper. Saute on high, tossing and moving them so they cook quickly, until the mushrooms have released their water and have started to brown, 3 to 5 minutes. Remove them to a plate and wipe the skillet carefully with a paper towel.
To assemble the pizza: Place the pizza crust in the skillet. Spread the pizza sauce over the crust leaving a 1/2-inch border around the edge. Sprinkle with the mozzarella. Imagine the pizza divided into four quadrants representing the seasons.
For spring: Arrange the artichoke hearts over one quadrant and sprinkle with the fontina.
For summer: Arrange the tomatoes on a second quadrant cut-side up, placing the mozzarella pearls around them.
For fall: Arrange the sauteed mushrooms on a third quadrant.
For winter: Arrange the pepperoni and olives on the last quadrant.
Transfer the skillet to the oven and bake until the pizza is hot and the cheese is melted, about 10 minutes. Garnish with the Parmesan and basil.