Set your oven rack in the upper third and preheat it to 375°F. Lightly butter a 9x13-inch broiler-safe baking dish.
Fill a large pot with water, heavily salt it, and bring to a rolling boil. Meanwhile, place each chicken breast between plastic wrap or inside a zip-top bag and gently pound to about 1/2-inch thickness. Season the chicken on all sides with half the thyme, 1 teaspoon salt, and black pepper.
Cook the penne pasta in the boiling water according to package instructions until just tender, about 9 to 11 minutes; then drain and return it to the pot.
Warm the olive oil in a skillet over medium-high heat until hot. Add the chicken and cover the pan; cook undisturbed for 4 to 5 minutes until the bottom turns golden. Flip the chicken, cover again, and cook for about 3 minutes longer until fully cooked through. Transfer chicken to a cutting board and let rest for at least 5 minutes.
Without washing the skillet, return it to medium heat. Melt butter while stirring and scraping any browned bits from the pan bottom. Add minced garlic and sauté, stirring frequently, until aromatic, roughly 1 minute. Stir in flour and cook continuously for about 30 seconds until lightly toasted.
Gradually pour in the milk and heavy cream while whisking steadily. Add the remaining salt and grated nutmeg. Bring this mixture to a gentle simmer, whisking occasionally. Meanwhile, chop the rested chicken into bite-size pieces.
Remove the pan from heat and gradually whisk in the Parmesan cheese until smooth and melted. Pour this sauce over the drained pasta and toss well to coat evenly.
Spoon half of the pasta mixture into the prepared baking dish and spread it evenly. Scatter half of the chicken pieces and half of the shredded mozzarella on top. Repeat with the remaining pasta, chicken, and mozzarella layers.
Bake the casserole uncovered until it bubbles along the edges, about 10 to 20 minutes. Switch the oven to broil and broil for 3 to 4 minutes until the cheese develops golden brown spots.
Allow the casserole to rest for 5 minutes before serving. If desired, sprinkle with fresh coarsely chopped thyme or parsley for garnish.