Prepare the chicken – Add the whole chicken, carrot, celery, salt, and black pepper in a large heavy-bottomed pot or Dutch oven. Cover the chicken with enough water to fully submerge it by at least 1-2 inches of water.
Cook until tender – Bring to a boil over medium-high heat, then reduce to a gentle simmer. Let the chicken simmer for 60-90 minutes until the meat is tender and easily pulls from the bone. Skim off any foam or impurities that rise to the top.
Remove the chicken and shred – Carefully transfer the chicken to a large bowl or plate. Let it cool, then shred the meat into bite-sized pieces. Discard the bones, skin, and vegetable scraps from the broth.
Strain the broth – Pour the broth through a fine-mesh strainer into a clean bowl or pot to remove any solids. Set aside at least 5-6 cups of strained broth for cooking the rice.
Brown the sausage – Heat one tablespoon of olive oil over medium heat in the same pot. Add the sliced andouille sausage and cook for 6-8 minutes, stirring occasionally, until browned and slightly crispy. Remove and set aside.
Sauté the onion – Lower the heat slightly and melt two tablespoons of butter in the pot. Add the diced onion and cook until softened (about 5-6 minutes).
Add the garlic and spices – Stir in the minced garlic, smoked paprika (optional), dried thyme, dried oregano (optional), and red pepper flakes (optional). Stir well to coat the onions in the seasonings.
Toast the rice – Add the rice to the pot and toast it, stirring frequently, until it turns lightly golden and smells toasty.
Pour in the Broth – Add 5 cups of reserved chicken broth and bay leaves. Stir well. Bring the mixture to a boil and return the shredded chicken and cooked sausage. Mix well. Note: Add more broth (up to one cup) if you like creamier, wetter, "boggier" rice.
Cook the rice – Return to a boil, then reduce the heat to low. Cover and simmer for 20-25 minutes until the rice is tender and the liquid is absorbed.
Let the rice rest – Allow the rice to rest, covered, for 5-10 minutes.
Season to taste - Stir in 1 teaspoon of lemon zest (or 1 tablespoon of fresh lemon juice) to create a fresh, bright contrast to the smoky flavors. Taste and adjust with extra salt or pepper if needed. Remove the bay leaves before serving.