Cook the spaghetti in a large pot of salted boiling water until al dente according to package directions; reserve 1/2 cup pasta cooking water, then drain.
In a large skillet over medium-low heat, melt the butter with the olive oil. Sprinkle in the flour and cook, stirring constantly, about 2 minutes until it forms a thick roux.
Add the minced garlic to the roux and cook 30 seconds more, stirring so it does not brown.
Turn the heat to low and slowly whisk in the chicken broth, then whisk in the heavy cream and the can of fire-roasted tomatoes with juices.
Stir in the chicken bouillon, dried parsley, dried basil, dried oregano, onion powder, salt, black pepper, and red pepper flakes. Increase heat to medium-high and simmer, stirring occasionally, 5–7 minutes until the sauce thickens.
Reduce heat to low and stir in the grated sharp cheddar until melted, then add the grated Parmesan and stir until incorporated.
Add the shredded rotisserie chicken and heat through, then add the cooked spaghetti and toss to coat, adding reserved pasta water a tablespoon at a time if the sauce is too thick.
Taste and adjust seasoning with additional salt, pepper, or red pepper flakes as desired. Garnish with extra Parmesan and parsley if desired, then serve.