Preheat oven to 375 degrees. Season chicken with salt & pepper and place on a foil lined baking sheet sprayed with non-stick cooking spray. Bake for about 20 minutes or until juices run clear.
While chicken is baking, in a 5-quart Dutch oven, make Penne pasta per package directions. Drain, cover, set aside to keep warm.
Add 3 tablespoons of butter to a large skillet and melt. Add onions, season with salt & pepper; cook for about 15 minutes, stirring frequently, until translucent. Add mushrooms and more salt & pepper. Add the minced garlic and Italian seasoning and gently stir until mushrooms have cooked down.
Once chicken has cooled to touch, carefully cut into cubes. Add the rest of the butter, milk, and mozzarella cheese to the pasta and stir. Add the chicken and onion mixture to the pasta and return the Dutch oven over the heat. Add the pesto sauce and stir gently until heated through.
Garnish with additional shredded cheese or freshly grated Parmesan cheese, if desired.