In a large Dutch oven or heavy pot, heat oil over medium-high. Add sausage and cook, stirring occasionally until brown all over, about 4 minutes. With a slotted spoon, transfer sausage to a dish. Season chicken on both sides with salt and pepper and add to pan, meaty side down. Cook until golden brown on both sides, 10 minutes, flipping once. Transfer chicken to dish with sausage.

Add onion, potatoes, and celery to pot and season with salt and pepper. Cook,stirring often, until onion is soft, 3 to 5 minutes. Return sausage and chicken, meaty side up, to pot and add 1 cup water. Bring to a boil, then cover and reduce heat to low. Simmer until potatoes are tender and chicken is cooked through, about 20 minutes.

Remove lid and increase heat to high; boil until liquid is reduced to 1/2 cup, 5 minutes. Transfer chicken to a platter. Add red wine vinegar and oregano to vegetables and sausage and stir to combine. With slotted spoon, transfer vegetables and sausage to platter with chicken and top with celery leaves. Transfer sauce to a gravy boat and serve alongside chicken.