Heat olive oil in a large deep skillet or Dutch oven over medium-high heat.
Add the chicken pieces and cook for 5-6 minutes, stirring occasionally, until the chicken is browned and cooked through.
Add minced garlic, dried Italian seasoning, salt, and black pepper to the skillet and sauté for 1 minute until fragrant.
Pour in the chicken broth and whole milk, stirring to combine.
Add the broken spaghetti to the skillet, pressing it down gently to submerge in the liquid.
Bring the mixture to a boil, then reduce heat to medium-low and cover the skillet with a lid.
Simmer for 12-15 minutes, stirring occasionally, until the spaghetti is tender and most of the liquid is absorbed.
Remove the lid and stir in the shredded cheddar cheese, mozzarella cheese, and grated Parmesan cheese until melted and creamy.
Cook uncovered for an additional 2-3 minutes, stirring frequently, until the sauce thickens slightly.
Remove from heat and let the dish rest for 2 minutes before serving.