Because andouille varies in spiciness, we suggest tasting a piece of the cooked sausage and then adjusting the amount of cayenne in the jambalaya to suit your taste. If you can’t find andouille, try tasso, chorizo, or linguiça; if using chorizo or linguiça, consider doubling the amount of cayenne. The onion, celery, bell pepper, and garlic can be chopped by hand instead of in the food processor. The shrimp don’t need to be deveined, but you can do so if you prefer. If you’re serving only four people, you may choose to skip the shredding step and serve the chicken on the bone.
In food processor, pulse onion, celery, red pepper, and garlic until chopped fine, about six 1-second pulses, scraping down sides of bowl once or twice. Do not overprocess; vegetables should not be pureed.
Heat oil in large heavy-bottomed Dutch oven over medium-high heat until shimmering but not smoking, about 2 minutes. Add chicken, skin-side down, and cook until golden brown, about 5 minutes. Using tongs, turn chicken and cook until golden brown on second side, about 3 minutes longer. Transfer chicken to plate and set aside. Reduce heat to medium and add andouille; cook, stirring frequently, until browned, about 3 minutes. Using slotted spoon, transfer sausage to paper towel–lined plate and set aside.
Reduce heat to medium-low, add vegetables, and cook, stirring occasionally and scraping bottom of pot with wooden spoon, until vegetables have softened, about 4 minutes. Add rice, salt, thyme, and cayenne; cook, stirring frequently, until rice is coated with fat, about 1 minute. Add tomatoes, reserved tomato juice, clam juice, chicken broth, bay leaves, and browned sausage to pot; stir to combine. Remove and discard skin from chicken; place chicken, skinned-side down, on rice. Bring to boil, reduce heat to low, cover, and simmer for 15 minutes. Stir once, keeping chicken on top, skinned-side down. Replace cover and continue to simmer until chicken is no longer pink when cut into with paring knife, about 10 minutes more; transfer chicken to clean plate and set aside. Scatter shrimp over rice, cover, and continue to cook until rice is fully tender and shrimp are opaque and cooked through, about 5 minutes more.
While shrimp are cooking, shred chicken (see illustration below). When shrimp are cooked, discard bay leaves; off heat, stir in parsley and shredded chicken, and serve immediately.