Adjust the rack to 6-8 inches below the broiler element and heat broiler to high.
Heat oil in a large cast iron skillet (11- or 12-inch) or stovetop griddle over high heat until oil shimmers. Reduce heat to medium.
Place tortilla in the skillet. Spread pizza sauce evenly over tortillas to the edges. Sprinkle with mozzarella and half the Parmesan cheese, making sure it overlaps the edges of the tortilla a bit. This will ensure a crispy, cheesy edge.
Add diced bell pepper and top tortilla with pepperoni, sausage chunks and mushrooms (if using). Cook until it is nice and toasty on the bottom.
Place skillet under the broiler and cook until the cheese is melted and starting to brown lightly (about 2-4 minutes).
Remove from the oven and sprinkle with remaining Parmesan and the basil. Using a small metal spatula, gently pry the edges of the pizza, releasing the cheese from the skillet. If more crispness is desired on the bottom of tortilla, place skillet over medium heat and cook until it is nicely toasted on the bottom.
Slide pizza out onto a cutting board. Sprinkle basil pieces over pizza. Cut and serve immediately.