How to Make It
Preheat the oven to 350°. Place the head of garlic, cut side up, on a double sheet of foil and drizzle with olive oil. Wrap the foil around the garlic and bake until the cloves are very soft, about 1 hour. Let cool. Squeeze the garlic onto a plate.
Roast the red pepper directly over a gas flame or under the broiler, turning frequently, until charred all over. Transfer the pepper to a plate to cool. Discard the charred skin, seeds and core and coarsely chop the pepper. In a food processor, puree the pepper with the roasted garlic, basil, oregano and the 2 tablespoons of olive oil. Season with salt and pepper.
On a lightly floured work surface, cut the pizza dough into 4 pieces and roll each piece into a ball. Flatten each ball, dust with flour and roll out to a 5-inch round. Cover with plastic wrap and let rest for 20 minutes.
Set a pizza stone on the bottom of the oven and preheat the oven to 500° for at least 45 minutes. Flour a pizza peel. Place a dough round on the peel and roll it out to an 1/8-inch thickness. Spread 2 tablespoons of the red pepper sauce over the dough to within 1/2 inch of the edge. Scatter 1/2 cup of the shredded chicken and one-fourth of the red pepper strips over the pizza, then top with 1/2 cup of the Manchego and 1/4 cup of the goat cheese. Slide the pizza onto the hot stone and bake until browned on the bottom and bubbling on top, about 6 minutes. Transfer to a cutting board, scatter some arugula leaves on top and cut into pieces. Repeat with the remaining dough and toppings.