Preheat oven to 375°F (190°C). Grease a baking dish with butter or oil.
Cook the pasta according to package instructions until al dente. Drain and set aside.
In a large pan, heat olive oil and butter over medium heat. Sauté garlic until fragrant.
Add chicken broth and heavy cream. Stir in oregano, salt, black pepper, and red pepper flakes.
Mix in shredded chicken and half of the Parmesan cheese. Let simmer for 2 minutes.
Toss cooked pasta with the creamy garlic sauce, ensuring even coating.
Transfer the pasta mixture to the baking dish. Sprinkle mozzarella and remaining Parmesan on top.
Bake for 20 minutes or until golden and bubbly.
Garnish with fresh parsley before serving.