Preheat your oven to 375 degrees Fahrenheit. Lightly grease a 9×13 inch baking dish.
Heat the olive oil in a large, oven-safe pot or Dutch oven over medium heat. Add the chopped onion and cook until soft, about 5 minutes. Add the minced garlic and cook for 1 minute more until fragrant.
Stir in the cream of chicken soup, cream of mushroom soup, chicken broth, Italian seasoning, salt, and pepper. Bring the mixture to a simmer.
Add the broken spaghetti pieces to the simmering sauce. Stir well to coat. Cover the pot and cook on low heat for 10 minutes, stirring halfway through to prevent sticking. The pasta should be mostly tender.
Remove the pot from the heat. Stir in the shredded chicken, milk, 1 1/2 cups of the cheddar cheese, and the Monterey Jack cheese until combined.
Transfer the mixture to the prepared baking dish if your pot is not oven-safe. If using an oven-safe Dutch oven, leave it as is.
Top the spaghetti evenly with the remaining 1/2 cup of cheddar cheese and the Parmesan cheese.
Bake uncovered for 15 to 20 minutes, or until the sauce is bubbly and the cheese topping is melted and lightly golden brown.
Let the cheesy baked spaghetti rest for 5 minutes before serving. Garnish with fresh parsley.
Notes
For a low-carb option, substitute the spaghetti with an equal amount of cooked spaghetti squash strands.
If you do not have leftover chicken, cook 1 pound of chicken breasts and shred them before starting the recipe.
You can substitute the cream of mushroom soup with another can of cream of chicken soup for a stronger chicken flavor.