Heat the oven to 375°F. Cook penne in water with the tablespoon salt until 2 minutes short of the package time, then drain.
Meanwhile, season chicken with 1/2 teaspoon salt and pepper. Heat oil in a large skillet and cook chicken for 7 to 8 minutes until at least 165°F; transfer to a plate.
Melt butter in the skillet, whisk in flour for 1 minute and gradually add milk and cream. Simmer gently for 3 minutes, then remove from heat and stir in Parmesan and remaining salt.
Fold in penne, chicken and frozen peas and transfer to a deep 9-by-13-inch baking dish. Scatter mozzarella over the top and cover with foil.
Bake for 25 minutes, then uncover for 5 to 8 minutes until bubbling and the center reaches 165°F. Rest for 5 minutes before serving.