In a heavy-bottom saucepan over medium-high heat, add the olive oil and chopped turkey necks, browning the bones on all sides, 6 to 7 minutes. Add the celery, carrot, garlic and onion and saute until the onions are translucent, 5 to 6 minutes. Season with salt and pepper.
Reduce the heat to medium, then add the flour to create a roux. Stir well and cook until the roux is a deep brown color and has a nutty aroma, 8 to 10 minutes. Add the chicken broth and thyme, while using a whisk to stir. Stir and cook until the gravy is smooth and there are no lumps. Bring to a boil then simmer to reduce slightly, about 10 minutes.
Remove the gravy from the heat and strain through a fine-mesh strainer into a smaller saucepan. Discard the solids. Taste and adjust seasoning. Keep warm until ready to serve.
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