Preheat the oven to 350°F. Lightly spray a 9-by-13-inch baking dish with non-stick cooking spray.
Melt 4 tablespoons butter in a large saucepan or deep skillet over medium heat. Add 1/2 cup chopped onion and 1 tablespoon minced garlic. Cook for about 3 minutes, then add 1 chopped yellow bell pepper and 1 chopped red bell pepper. Cook until the peppers start to soften.
Add 1 cup sliced baby Bella mushrooms. Cook for a few minutes, stirring occasionally, until softened and most of the moisture has cooked away.
Add 1 1/2 cups heavy cream and 1/2 cup chicken broth. Stir in 2 cups grated white cheddar, 8 ounces cream cheese, 2 ounces Velveeta, and 1 teaspoon salt.
Reduce the heat to medium-low and stir until the cheeses melt and the sauce is smooth. Keep it at a gentle simmer.
Cook the spaghetti according to the package directions until tender but not overly soft. Drain well.
Pull apart 1/2 rotisserie chicken, checking for bones. Stir the chicken into the warm cheese sauce and add black pepper to taste. Taste before adding more salt.
Add the drained spaghetti to the sauce. Gently toss until the noodles are coated and the chicken and vegetables are evenly mixed in.
Transfer the mixture to the prepared baking dish and spread evenly. Sprinkle shredded cheddar over the top.
Cover with foil and bake at 350°F for 20 minutes. Remove the foil and bake for another 5 to 10 minutes, until the cheese melts and the top is lightly browned.
Let the casserole rest for a few minutes. Sprinkle with fresh parsley and serve warm, scooping the spaghetti so each serving gets chicken, vegetables, sauce, and melted cheese.