Heat oven to 450°F and roast poblanos on a sheet pan for 20 to 25 minutes, turning once, until blistered. Cover in a bowl for 10 minutes, then peel, seed and chop finely.
Heat oil in a skillet over medium heat and soften onion for 5 minutes. Add ground chicken and cook for 8 to 10 minutes, breaking it into fine crumbles, until 165°F; stir in garlic and cumin for 1 minute.
Transfer chicken mixture and chopped poblanos to a 6-quart slow cooker. Add salt, pepper and the 3 1/2 cups broth, cover and cook on low for 3 hours.
Turn to high and stir in milk and corn. Add dry spaghetti in several crossed layers, press gently into the liquid and cover for 15 minutes.
Stir carefully to separate the softening strands, cover and cook for 15 to 25 minutes more until tender, stirring once halfway. Add some of the measured extra broth in 1/4-cup amounts only if the strands are still firm and the liquid is nearly absorbed.
Switch off and stir in cream cheese until smooth, then add cheddar a handful at a time. Rest for 3 minutes and divide among six bowls, serving before the spaghetti absorbs more sauce.