Add 1 tablespoon olive oil to a large high-sided skillet or Dutch oven set over medium heat. Add the chopped red onion, minced garlic, sliced celery, and carrot half-rounds. Season with salt and black pepper to taste. Stir to combine and cook, stirring frequently, until the vegetables soften and develop some color, approximately 4-5 minutes. You want them to be tender and fragrant, with the onions turning translucent.

Add the chicken thighs (whole), sliced andouille sausage, chopped rosemary, thyme leaves, and bay leaves to the pot. Season with additional salt and black pepper if desired. Pour in the 3 cups of chicken broth and stir gently to combine, making sure the chicken is mostly submerged.

Increase the heat to high and bring the mixture to a boil. As soon as it reaches a boil, immediately reduce the heat to medium-low. Cover the pot with a lid and let it simmer for 45-50 minutes, or until the chicken pulls apart easily with two forks. The chicken should be incredibly tender and the broth should be rich and flavorful.

Add the drained and rinsed cannellini beans to the pot and stir to combine. Let the stew simmer for another 4-5 minutes, just until the beans are heated through. Remove the pot from heat, fish out the bay leaves, and stir in the fresh chopped parsley right before serving.