Put the sausage and oil in a large, cold nonstick skillet and cook over medium heat until the fat starts to render and the sausage starts to brown, about 6 minutes. Transfer the sausage to a small plate with a slotted spoon.
Sprinkle the chicken with 3/4 teaspoon salt and a couple turns of pepper, then toss in the flour to coat. Increase the heat under the skillet to medium-high and sear the chicken skin-side down until golden brown, about 8 minutes.
Flip the chicken skin-side up. Add the scallion whites, garlic, jalapeno, allspice, thyme and 3/4 cup water. Return the sausage to the pan. Arrange the celery around the chicken and bring to a simmer. Reduce the heat to medium-low, cover and cook until the chicken is cooked through and the celery is crisp-tender, 20 to 25 minutes. Serve topped with the scallion greens and a few sprigs of fresh thyme.
Copyright 2016 Television Food Network, G.P. All rights reserved.