Preheat your oven to 350°F and grease a 9×13-inch baking dish with butter or non-stick spray.
Bring a large pot of salted water to a boil on your stovetop. Cook the spaghetti noodles according to the package directions, but strain them 2 minutes early so they remain highly al dente.
While the pasta cooks, melt 1 tablespoon of butter in a skillet over medium heat. Toss in the chopped celery, onions, and bell peppers. Sauté for 4 to 5 minutes until the onions turn translucent and soft. Pro Tip: Sweating your aromatics in butter before they enter the casserole is crucial. Raw vegetables release a large amount of internal water as they bake, which can easily dilute your cream sauce and make the entire dish soggy. Sautéing them first cooks off that excess water while concentrating their natural sugars.
In a large mixing bowl, combine the shredded chicken, the par-boiled spaghetti noodles, and your warm sautéed vegetable mix.
In a separate medium bowl, vigorously whisk together the chicken broth, the 2 cans of condensed cream of chicken soup, and a pinch of salt and black pepper until completely smooth and unified. Crucial: Whisking the liquid components completely smooth in a separate bowl creates an emulsified sauce shield. Dumping cold, thick canned soup directly onto dry pasta makes it impossible to distribute evenly, leaving you with pockets of dry noodles and dense clumps of soup.
Pour the whisked soup mixture directly over the large bowl of chicken and noodles. Use a large spoon or kitchen tongs to thoroughly toss the ingredients until every strand of spaghetti is deeply coated in the sauce.
Spread the mixture evenly into your prepared 9×13-inch baking dish. Blanket the top surface uniformly with the freshly shredded cheddar cheese.
Cover the dish tightly with aluminum foil and bake for 35 minutes. Carefully slide the foil off and bake uncovered for an additional 10 minutes until the cheese is entirely liquefied, bubbling at the edges, and developing small golden spots. Let the casserole rest for 10 minutes on a wire rack to allow the internal starches to set before scooping.