Heat the oven to 350°F and set a rack in the center. Rub the inside of a 9×13-inch baking dish with the butter wrapper. Spray works too, but the wrapper leaves a thicker film on the glass.
Boil the spaghetti in well-salted water for 9 minutes, 2 minutes under the box time. Drain it and do not rinse. The center should still feel firm when you bite a strand. Soft noodles turn mushy once they bake in the sauce.
Melt the butter in a Dutch oven over medium heat. Cook the onion and both peppers for 6 minutes. Add the garlic for 40 seconds, then the mushrooms for 2 minutes. The onion should look see-through. One evening the garlic scorched while I answered a call.
Burned garlic tastes bitter, so I scooped it out and added fresh cloves.
Lower the heat to medium-low. Pour in the dairy and broth, then add the cream cheese and Velveeta. Whisk for 5 minutes until no lumps remain.
Keep the pot below a boil. A greasy, grainy sauce means the heat ran too high. Pull the pot off the burner and whisk in 2 tablespoons of cold milk.
Stir in 2 cups of the cheddar, a handful at a time. Add the salt and pepper, then taste the sauce.
It should taste a little bold on its own. The chicken and pasta soak up seasoning, so a sauce that seems right now will taste flat later.
Add the chicken, then the spaghetti. Toss with tongs until every strand is coated.
The pot will look crowded but loose. Sauce should pool a little at the bottom when you lift the noodles.
Pour everything into the dish and twirl strands into nests with tongs. Sprinkle on the last cup of cheddar and cover with foil. Bake 22 minutes, uncover, and bake 8 minutes more.
Look for bubbling edges and golden, lacy spots on top. Rest the dish 10 minutes, then scatter the parsley.