In a large frying pan with a lid, heat the oil over medium heat until it shimmers and slides across the pan. Add the onions and cook until translucent, about 5 minutes.
Slice the chicken breasts thinly and add to the pan with the onions. Add the garlic, 1/2 cup of the hot sauce and 1 cup water. Bring to a simmer, then cover the pan and reduce the heat to low. Cook until the chicken is falling apart and the sauce has reduced to almost dry, 45 minutes to 1 hour.
Combine the sour cream, mayonnaise, Cheddar and 1/4 cup of the blue cheese crumbles in a bowl and stir until well combined. Transfer the dip to a serving bowl. Stir in the remaining 2 tablespoons hot sauce, and top with the green onions and the remaining 1/4 cup blue cheese crumbles.
Serve immediately, with the chicken, celery, carrots, chips and other crudites.