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 in the olive oil. Whisk in the flour and cook, stirring, for about 2 minutes until the mixture thickens and is lightly bubbly.
Add the minced garlic and cook, stirring, for 30 seconds more.
Reduce heat to low and slowly whisk in the chicken broth, then the heavy cream, followed by the fire roasted tomatoes with their juices and the chicken bouillon; add all dried seasonings.
Increase heat to medium-high and bring the sauce to a simmer, stirring constantly; simmer 5–7 minutes until slightly thickened.
Lower the heat and stir in the cheddar until melted, then stir in the Parmesan until smooth.
Stir in the shredded chicken and heat through, then add the cooked spaghetti and toss to coat; add reserved pasta water a little at a time if the sauce needs thinning.
Taste and adjust seasoning with additional salt, pepper, or red pepper flakes as desired, and garnish with extra Parmesan and parsley if you like.
