In a cast-iron skillet, heat the olive oil over medium-high heat. Sprinkle the chicken evenly on both sides with the salt and pepper. Add the chicken to the pan and sear until golden on both sides, 2 or 3 minutes per side. Place the whole skillet in the oven and bake until the chicken is cooked through, about 15 minutes. Transfer the chicken to a bowl to cool slightly, then shred the chicken.
Increase the oven temperature to 500°F. Place a cast-iron pizza stone in the oven to preheat for 15 to 20 minutes.
When the dough has risen, remove the plastic wrap and push down on the dough to release the gases and bubbles. Shape it into a ball. Stretch the pizza dough into a 14-inch round on a sheet of parchment. (If the dough is sticky, dust it lightly with flour.) Spread the pesto over the dough, then layer on the chicken, bell peppers, and shredded cheeses. Transfer the pizza and parchment to the hot pizza stone.
Bake until the crust is golden and the topping is bubbling, 13 to 15 minutes. When the pizza comes out, mix the melted butter and garlic powder and brush it along the crust of the pizza.
Transfer to a cutting board and cut into 8 slices to serve.
Homemade Pizza Dough
In a small bowl, stir together the yeast, sugar, and warm water. Let stand until foamy, about 5 minutes.
Add the flour to the bowl of a stand mixer fitted with the dough hook. Sprinkle the salt over the flour and mix on low speed for 15 seconds to distribute the salt.
Add the olive oil, room-temperature water, and yeast mixture and mix on low speed for 1 to 2 minutes, then increase the speed to medium and mix until a ball of dough forms, about 5 minutes.
Spray a medium bowl lightly with cooking spray. Shape the dough into a ball and set it in the bowl seam side down. Cover the bowl with plastic wrap and place in a warm area until doubled in size, about 1 hour.
When the dough is almost risen, place a cast-iron pizza stone on the center rack of the oven and preheat the oven to 500°F.
When the dough has risen, remove the plastic wrap and push down on the dough to release the gases and bubbles. Shape it into a ball. (If using the dough later, cover with plastic wrap and place in the refrigerator for up to 12 hours. The dough will slowly proof during that time. Once you’re ready to use it, set it out for 30 minutes to come to room temperature.)
Spread a piece of parchment paper about 1 1/2-feet long on a work surface. Stretch the dough into a 14-inch round on the parchment.
Top with your favorite toppings. Carefully pull the oven rack with the pizza stone halfway out of the oven and transfer the parchment and pizza dough to the stone.
Bake until the crust is golden, 13 to 15 minutes. Transfer the pizza to a cutting board and cut into 8 slices to serve.
In a food processor, combine the basil, garlic, and pine nuts. Pulse until roughly chopped, 10 to 15 seconds.
With the machine running, slowly drizzle in the olive oil until well combined, 1 to 2 minutes.
Add the Parmesan, salt, and pepper and mix until just incorporated.
Serve immediately, or store in an airtight container in the refrigerator for up to 3 days or in the freezer for up to 1 month.