Season the turkey wings with salt and black pepper.
Heat vegetable oil or butter in a large skillet or Dutch oven over medium-high heat and brown the wings on all sides until golden, about 5-7 minutes.
Add sliced onion, carrots, and celery if using, and cook until softened, about 5 minutes.
Stir in minced garlic and cook for 1 more minute.
Pour in the broth and add dried thyme or poultry seasoning, then bring to a simmer.
Reduce heat to low, cover, and let the wings simmer gently for 45-60 minutes until tender and near falling off the bone.
Remove the turkey wings and keep warm. Whisk the flour into the simmering liquid in the pan until smooth.
Continue to cook the gravy until thickened to desired consistency, adjusting salt and pepper as needed.
Return the wings to the gravy or pour gravy over the wings on a serving dish.
Garnish with chopped parsley if desired and serve with mashed potatoes, rice, or biscuits.