Preheat oven to 375°F.
Carefully fillet each chicken breast into two thinner cutlets.
Heat olive oil in a large deep skillet over medium heat.
Add chicken cutlets and cook until golden brown and fully cooked, about 7–10 minutes total.
Transfer chicken to a cutting board and let rest.
Bring a large pot of water to a boil. Add salt and cook spaghetti until tender or al dente. Drain well.
In the same skillet, melt butter over medium-high heat.
Add diced onion and minced garlic. Sauté for 3–4 minutes until softened.
Sprinkle flour into the skillet and stir continuously for 2–3 minutes until lightly golden.
Slowly whisk in chicken broth until smooth.
Add salt, black pepper, Italian seasoning, heavy cream, 1/2 cup cheddar cheese, and 1/2 cup mozzarella cheese.
Stir until melted and creamy.
Chop the cooked chicken into bite-sized pieces.
Turn off the heat and add drained tomatoes with green chiles, cooked chicken, and spaghetti.
Toss everything together until evenly coated.
Transfer the Chicken Spaghetti mixture into a 9×13-inch casserole dish.
Top with remaining cheddar and mozzarella cheese.
Bake for 25 minutes until bubbly and golden.
Let rest for 5 minutes before serving.