Bring a large pot of salted water to a boil and cook the spaghetti until al dente according to package directions; reserve 1/2 cup pasta water, then drain and set aside.
In a large skillet over medium-low heat, melt the butter in the olive oil. Sprinkle in the flour and stir constantly for about 2 minutes until the mixture is thick and slightly cooked.
Add the minced garlic to the roux and cook for 30 seconds, stirring so the garlic does not brown.
Lower heat to low and slowly whisk in the chicken broth until smooth, then whisk in the heavy cream, canned fire-roasted tomatoes with their juices, and chicken bouillon.
Add the dried parsley, dried basil, oregano, onion powder, salt, black pepper, and red pepper flakes; increase heat to medium-high and bring the sauce to a simmer, stirring constantly, then simmer 5–7 minutes until slightly thickened.
Reduce heat to low and stir in the grated cheddar until melted, then stir in the Parmesan until melted and the sauce is smooth.
Stir the shredded rotisserie chicken into the sauce and heat through.
Add the cooked spaghetti to the skillet and toss to coat, using reserved pasta water a little at a time to loosen the sauce if needed; taste and adjust seasoning with salt, pepper, or extra red pepper flakes.
If desired, transfer to a lightly greased 9×13 baking dish, top with an extra 1 cup shredded cheddar, and bake covered at 350°F for 25–35 minutes until heated through and cheese is melted.
Garnish with additional grated Parmesan and parsley if desired, then serve.