Heat the oven to 375°F. Bring broth to a gentle simmer in a saucepan, add chicken and poach for 10 to 15 minutes until the thickest meat reaches 165°F. Remove and shred; measure 2 cups hot poaching broth for the sauce.
Meanwhile, cook spaghetti in water with the tablespoon salt until 2 minutes short of the package time and drain. Melt butter in a large skillet and cook onion, bell pepper and celery for 6 minutes.
Stir flour into the vegetables for 1 minute, then gradually add the measured broth and milk. Simmer for 3 minutes, stir in tomatoes with chiles and their juices, and remove from the heat.
Melt 4 oz cheddar into the sauce and add sauce salt and pepper. Fold in shredded chicken and spaghetti, transfer to a deep 9-by-13-inch baking dish and scatter remaining cheddar over the top.
Cover and bake for 20 minutes, then uncover for 8 to 10 minutes until bubbling and the center reaches 165°F. Rest for 5 minutes before serving.