If using rotisserie chicken, remove the skin and pull the meat off the bones, then shred it with two forks until you have about 3 cups. If poaching chicken, place 1 1/4 pounds boneless, skinless chicken breasts in a saucepan, cover with water by 1 inch, add a pinch of salt, and simmer for 15 to 18 minutes until the thickest part reaches 165°F; rest 5 minutes, then shred.
Finely dice the onion, celery, and carrots to about pea size, mince the garlic, shred the cheddar on the large holes of a box grater, and cube the cream cheese so it softens.
Melt the butter in a Dutch oven over medium heat, add the onion, celery, and carrots, and cook for 6 to 8 minutes until soft and translucent. Add the garlic and stir for 30 seconds until fragrant.
Sprinkle the flour and ranch seasoning over the vegetables and stir constantly for 2 minutes so the flour cooks and loses its raw taste; the mixture will look thick and pasty.
Pour in the chicken broth slowly, about 1 cup at a time, stirring well after each addition so no lumps form, then stir in the half-and-half, Buffalo wing sauce, kosher salt, and black pepper.
Bring the soup to a gentle simmer over medium heat, then reduce to medium-low and simmer for 8 to 10 minutes, stirring occasionally, until slightly thickened; avoid a hard boil so the dairy doesn't separate.
Reduce the heat to low, add the softened cream cheese, and stir until melted and smooth, about 2 to 3 minutes. Add the shredded cheddar a handful at a time, stirring until each handful melts before adding the next.
Stir in the shredded chicken and heat for 4 to 5 minutes until warmed through. Taste and adjust with more salt, pepper, or Buffalo sauce, and thin with a splash of broth or half-and-half if needed.
Ladle into warm bowls and top each with blue cheese crumbles, sliced green onions, and a few celery leaves. Drizzle with ranch dressing if you like and serve right away.