Heat 1 tablespoon olive oil in a large sauté pan over medium heat.
Add the diced onion and cook, stirring occasionally, about 4 minutes until softened.
Add the minced garlic, the 1/4 cup oil-packed sun-dried tomatoes (chopped), and 1 teaspoon kosher salt; sauté about 1 minute until fragrant.
Add the cooked, shredded chicken and the cooked pasta (1 pound chicken; 1 16-ounce package pasta). Add the remaining 1 tablespoon olive oil and stir gently to combine. Cook 2–3 minutes, stirring occasionally, until the chicken and pasta are heated through.
Remove the pan from the heat. Add the 2 tablespoons chopped fresh basil, 1/2 cup shredded mozzarella, and 1/4 cup grated Parmesan; stir until the cheeses begin to melt and everything is evenly combined.
Serve immediately.