Pat chicken pieces dry with paper towels and season all over with 1 teaspoon kosher salt and 1 teaspoon black pepper. Roughly chop the onion and celery into large chunks, and smash the garlic cloves with the flat side of your knife.
Place the seasoned chicken in a large pot with the onion, celery, garlic, bay leaves, and thyme. Pour in 6 cups of water, bring to a boil over medium-high heat, skim off any foam, then reduce to medium-low, cover, and simmer for 45 to 50 minutes until the chicken is completely tender and falling off the bone.
Remove the chicken pieces with tongs and let cool for 10 minutes, then pull the meat from the bones and skin, shredding it into bite-sized pieces. Discard the skin and bones.
Pour the broth through a fine mesh strainer into a large bowl, discarding the spent aromatics. Measure out 4 cups of the strained broth to use for cooking the rice.
Return the pot to medium heat, add the butter, and let it melt. Add the sliced sausage in a single layer and brown for 2 to 3 minutes per side until nicely caramelized.
Pour the 4 cups of strained broth into the pot with the sausage and bring to a full boil. Taste and adjust seasoning, then add the rice and shredded chicken, stirring to combine everything evenly.
Bring back to a boil, reduce heat to low, cover, and cook for 20 minutes. Stir once or twice in the first 5 minutes, then keep the lid on and do not lift it. Turn off heat and rest, covered, for 10 minutes.
Stir the bog gently; the rice should be thick and sticky. If too wet, cook uncovered over medium-low heat for 3 to 5 minutes. Garnish with fresh parsley and sliced green onions and serve with hot sauce on the side.