Over medium-high heat, heat a dutch oven with 1 tablespoon of canola oil. Add the oxtails and brown all sides, about 5 minutes per side. Place each browned oxtail onto a plate. If you need to do this in batches, add additional canola oil per batch.
Once all of the oxtails have been browned, add 1 tablespoon of canola oil and brown the chicken wings. Transfer to a plate once browned.
Add 2 tablespoons of canola oil. Add the carrot, celery, shallots, and garlic and cook until slightly softened, about 5 minutes. Add the chicken broth and water. Add the oxtail and chicken wings. Finally add the thyme, rosemary, sage, bay leaves, dried shiitake mushrooms, and whole black peppercorns.
Bring the mixture to a boil and then reduce the heat. Keep the broth at a simmer, uncovered for at least 3 ½ hours. When the time has passed, taste for flavor to see if the broth has been simmering for long enough. Season with salt.
Once the broth is ready, strain all the pieces out and set the broth aside until you are ready to turn it into gravy.
To prepare the gravy, place a pot over medium-high heat. Melt the butter in the pan. Once the butter has melted, slowly add the flour while constantly whisking.
After the butter and flour have combined into a roux, slowly add the broth while whisking. It is very important that you do this slowly and to be constantly whisking. This will allow the gravy to become nice and thick.
Bring to a boil and then reduce the heat. Simmer until the gravy has thickened. Season with black pepper and salt to your liking.
