1 Brown the oxtails: Heat 2 tablespoons olive oil in a 5 to 6-quart thick bottomed Dutch oven on medium high to high heat. Working in batches, pat dry the oxtails with paper towels, sprinkle them on all sides with salt, and add them to the pan, fat side down on the pan. Add more oil as needed with additional batches of oxtails. Do not crowd the pan.
Let them get well browned on one side before using tongs to move them. Brown well on all sides. Remove to a large bowl.
2 Sauté onions, celery, carrots: Add the onions, celery and carrots to the pot. Sauté until translucent and lightly browned, about 5 minutes. Remove the vegetables from the pot to a bowl, cover and set aside.
3 Deglaze pan with wine, reduce: Add the bottle of wine to the pot. Increase the heat to as high as it will go, scraping up any browned bits from the bottom of the pan. Boil the wine, uncovered, until it is reduced to about a cup.
4 Add oxtails, stock, water, thyme, simmer 3 hours: Return the oxtails to the pot (but not the vegetables). Add the stock and enough water to just cover them. Add the thyme. Bring to a boil and reduce to a simmer, covered.
Simmer on the stovetop for 3 hours. (You can also place the simmering oxtails into a 350°F oven for the same amount of time if the oven is more convenient.)
5 Add the vegetables back to the pot when you have about a half hour left to go.
6 Chill: Remove from heat and let cool. Chill in the refrigerator overnight so that the flavors blend and the fat on the surface solidifies, making it easier to remove. You can skip this step, but the oxtails will be better if they are chilled in this state overnight.
7 Remove excess fat: The next day, remove the pot from the refrigerator and scrape off the layer of rendered fat that has solidified on top of the oxtails. If you are not waiting for the oxtails to chill, the fat still needs to be removed. If working with a room temperature or warm pot, use a fat separator or a large metal spoon to skim away the fat.
8 Reheat, remove meat from bones: Heat the oxtails on medium heat. Cook uncovered for about another half an hour, or until the meat can easily be pulled off the bones. Then use a slotted spoon to remove the oxtails from the pot. Let cool enough to touch. Use your hands to remove the meat from the bones to a bowl. Take care to remove as well the round tough cartilage caps on either end of the vertebrae.
9 Strain pot liquids, discard solids, reduce sauce: (Optional, if you want a smoother glaze) While the oxtails are cooling in the step above, strain the mixture in the pot, discarding the solids and returning the liquid to the pot. Increase the heat to high to bring the liquid to a boil.
10 Return oxtails to pot, simmer until sauce reduces to a glaze: When the liquid has reduced by about a half, add the oxtail meat back to the pot. Bring to a boil, continue to boil away the liquid until it has reduced to a light syrupy consistency.
As the mixture boils down, you may want to reduce the heat to a simmer, and stir the oxtails a little so that the glaze doesn't burn and so that the meat doesn't stick to the pan. When the right consistency, remove from heat and serve.
Serve with mashed potatoes, polenta, egg noodles, or rice. (Avoid egg noodles if cooking gluten-free.)