For the giblet stock: Heat the oil in a 6 1/2-quart stockpot over medium heat until it shimmers. Add the neck and backbone and saute until browned, 5 to 6 minutes. Add the giblets, onion, carrot, celery, and kosher salt. Cook, stirring occasionally, until vegetables are softened, 4 to 5 minutes. Pour in the water and add the thyme, rosemary, bay leaf, and black peppercorns. Cover and bring to a boil over high heat. Uncover, reduce heat to low, and simmer until the stock reduces to 3 cups, about 1 1/2 hours.
Strain the stock through a fine-mesh strainer and cool slightly. Discard all solids except for the giblets. Chop the giblets and set aside.
For the giblet gravy: Place 2 cups of the giblet stock in a 4-quart saucier over medium heat.
Combine 1/2 cup of cool stock with the all-purpose flour in a jar with a tightly fitted lid. Shake vigorously until starch dissolves and no lumps remain.
Gradually add the flour slurry to stock, whisking constantly. Bring to a boil and cook until thickened slightly, 3 to 4 minutes.
Combine the remaining 1/2 cup of cool stock with the potato starch in the lidded jar and shake vigorously until the starch dissolves and no lumps remain.
Turn off the heat, remove the saucier from the burner, and whisk until the liquid temperature decreases to 190ºF. Return to low heat and whisk in the potato starch slurry, rosemary, sage, thyme, salt, and pepper. Simmer for about 5 minutes to heat through and thicken, then season with additional salt and pepper, if desired. Add the chopped giblets and stir to combine. Keep warm in a thermos until ready to serve.