Set one corn cob upright in a medium bowl and carefully shave kernels off by slicing down the sides with a sharp, nearly vertical knife. Scrape the cob to release the corn milk into the bowl; reserve cobs.
Place sausage in a cold large skillet and cook over medium-high heat, breaking it up with a wooden spoon until fat is rendered, about 8 minutes. Transfer sausage to a medium bowl.
In the same skillet, heat olive oil over medium heat. Add onion, celery, green and red bell peppers, and serrano chile. Season with half the salt and black pepper. Cook, stirring frequently, until vegetables soften and turn translucent, about 5 minutes.
Add smoked paprika and cayenne pepper, stirring to coat vegetables and bloom spices for about 1 minute.
Add corn kernels with their milk, reserved cobs, and chicken broth. Cover and cook until cobs release some cream, about 5 minutes. Remove and discard cobs.
Stir in butter, remaining salt, and a few grinds of black pepper. Return sausage to skillet and combine all ingredients well.
Serve Maque Choux warm over cooked white rice or grits.