Heat 1 tablespoon olive oil in a large skillet over medium heat.
Add 1/2 chopped onion, 2 finely chopped carrots, and 2 ribs finely chopped celery; sprinkle with salt and sauté until softened, about 3–5 minutes.
Stir in 2 minced garlic cloves and cook for 1 minute more.
Add 1 pound ground chicken and break it up with a spoon; cook until no longer pink and lightly browned.
Pour in 8 ounces tomato sauce and 1/3 cup hot sauce, reduce heat to low, cover, and simmer for 5 minutes.
Uncover and stir, then continue to cook uncovered for 2–4 minutes until the sauce reduces and thickens; if too dry, stir in up to 1/4 cup water or chicken broth.
Spoon the buffalo chicken mixture onto the 4 hamburger buns and top with crumbled blue cheese, a drizzle of ranch dressing, and chopped chives as desired.