In a medium pan, heat the oil over medium-high heat. Add the onion, carrot, celery, and garlic. Cook until softened, about 5 minutes. Increase the heat to high and add the chicken. Cook, breaking it up with a spoon, until cooked through, 5-6 minutes. Season with salt and pepper.
In a small bowl, whisk together the hot sauce, water, tomato paste, brown sugar, chili powder, and cornstarch. Pour into the pan and bring to a simmer over medium heat. Cook 5-6 minutes, stirring frequently, until the sauce is reduced.
Divide the chicken among the rolls. Top each sandwich with 1 Tablespoon of Ranch dip.