Season the chicken liberally with salt and pepper.
Heat 1 tablespoon oil in a large Dutch oven over med-high heat until shimmering but not smoking, about 2 minutes.
Add five chicken thighs, skin side down, and cook, not moving them until the skin is crisp and well browned, about 5 minutes.
Using tongs, flip the chicken and brown on the second side, about 5 minutes longer.
Transfer the browned chicken to a large plate.
Brown the remaining 5 chicken thighs, transfer them to the plate, and set aside.
Drain the fat from the pan and strain it through a fine-mesh strainer lined with cheesecloth and into a measuring cup.
The remaining oil should be entirely free of particles.
Add enough vegetable oil (about 1/4 cup) to equal 1/2 c total fat.
Heat the chicken fat-vegetable oil mixture in a clean Dutch oven over med-high heat until it registers 200° on an instant-read thermometer, about 1 1/2-2 minutes.
Decrease the heat to medium and gradually stir in the flour with a wooden spoon or spatula, working out any lumps that form.
Continue stirring constantly, reaching into the edges of the pan, until the mixture has a toasty aroma and is a deep reddish brown, about the color of an old copper penny or between the colors of milk chocolate and dark chocolate, about 20 minutes (the roux will become thinner as it cooks; if it begins to smoke, remove the pan from the heat and stir the roux constantly to cool slightly).
Add the onions, bell pepper, celery, garlic, thyme, 1 teaspoon salt, and 1/4 teaspoon cayenne to the roux and cook, stirring frequently, until the vegetables soften, 8-10 minutes.
Add the chicken broth in a slow, steady stream while vigorously stirring.
Stir in 2 cups water and the bay leaves and place the browned chicken thighs in a single layer in the pot.
Increase the heat to high and bring to a boil.
Decrease the heat to med-low and skim off any foam that rises to the surface; simmer for 30 minutes.
Stir in the sausage and continue simmering to blend the flavors, about 30 minutes longer.
Off the heat, stir in the parsley and scallions and adjust the seasonings with salt, pepper, and cayenne to taste.