Arrange a rack in the top third of the oven and heat the oven to 375°F. Coat a 9x13-inch broiler-safe baking dish with butter.

Bring a large pot of heavily salted water to a boil over high heat. Meanwhile, pound out 2 medium boneless, skinless chicken breasts one at a time: Place the chicken breast between 2 sheets of plastic wrap or in a large zip-top bag. Pound with a rolling pin or flat side of a meat mallet to an even ½-inch thickness. Season the chicken all over with 1 teaspoon coarsely chopped fresh thyme leaves, 1 teaspoon of the kosher salt, and ½ teaspoon black pepper.

Add 1 pound dried penne pasta to the boiling water and cook according to the package directions until just al dente, 9 to 11 minutes. Drain and return the pasta to the pot. While the pasta is cooking, cook the chicken.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat until shimmering. Add the chicken. Cover and cook undisturbed until golden-brown on the bottom, 4 to 5 minutes. Flip the chicken, cover, and cook until the chicken is just cooked through, about 3 minutes more. Transfer to a clean cutting board and let rest for at least 5 minutes.

Meanwhile, return the skillet (do not clean) to medium heat. Add 3 tablespoons unsalted butter and melt, stirring and scraping up the browned bits from the bottom of the pan. Add 5 minced garlic cloves and cook, stirring often, until fragrant, about 1 minute. Stir in 3 tablespoons all-purpose flour and cook, stirring constantly, until lightly toasted, about 30 seconds more.

While whisking constantly, slowly pour in 3 cups whole milk and 1 cup heavy cream. Stir in the remaining 1 teaspoon kosher salt and 1 small pinch freshly grated nutmeg. Bring to a simmer; whisk occasionally. Meanwhile, cut the chicken into bite-size pieces.

Remove the skillet from the heat. Whisk in 5 ounces finely grated Parmesan cheese a handful at a time until smooth and fully melted. Pour into the pot with the drained pasta and toss to coat.

Transfer half of the pasta mixture into the baking dish (about 4 cups) and spread into an even layer. Scatter half of the chicken and 3 ounces of the shredded mozzarella cheese evenly over the pasta. Repeat layering the remaining pasta mixture and any sauce, chicken, and 3 ounces shredded mozzarella.

Bake uncovered until bubbling around the edges, 10 to 20 minutes. Turn the oven onto broil and broil until the cheese is browned in spots, 3 to 4 minutes. Let the casserole cool for 5 minutes. Garnish with coarsely chopped fresh thyme leaves if desired.