Preheat the oven to 475°F (245°C). Place a pizza stone or inverted baking sheet in the oven to heat.
Prepare the pesto: In a food processor, combine basil leaves, grated Parmesan cheese, pine nuts, and minced garlic. Pulse until finely chopped.
With the processor running, slowly drizzle in 1/3 cup olive oil until the mixture is smooth and emulsified. Season with 1/4 teaspoon salt and 1/4 teaspoon black pepper. Set aside.
On a lightly floured surface, roll out the pizza dough into a 12-inch circle.
Spread the prepared pesto evenly over the pizza dough, leaving a small border around the edges.
Distribute the shredded cooked chicken evenly over the pesto layer.
Sprinkle the shredded mozzarella cheese evenly over the pizza.
Carefully transfer the pizza onto the preheated pizza stone or baking sheet.
Bake for 12 to 15 minutes, or until the crust is golden and the cheese is melted and bubbly.
While the pizza bakes, prepare the arugula topping: In a small bowl, toss the fresh arugula with lemon juice, 1 tablespoon olive oil, 1/4 teaspoon salt, and 1/4 teaspoon black pepper.
Remove the pizza from the oven and immediately top with the dressed arugula.
Optionally, sprinkle additional grated Parmesan cheese over the top.
Slice and serve warm.