Clean and trim the oxtails of any excessive fat. Season generously with salt and pepper.
Add olive oil to a hot dutch oven, or pan deep enough to hold the meat and braising liquids. Sear the oxtails on all sides until evenly browned. Do this in batches if you need to.
Add the 2 halves of garlic to the pan, cut side down. Push them to the bottom of the pan.
Pour in the wine to deglaze the pan scraping up any bits stuck to the pan. Bring to a boil then reduce the heat to a simmer until the liquid is reduced by half. Then add the chicken stock and bring to a boil again.
Cover the pan and bake in the over at 325° for 2 1/2 - 3 hours.
Add olive oil to a hot pan and saute the onions and mushrooms until tender. Salt and pepper to tase. Add over top of the oxtails and serve.
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Notes
When adding your braising liquids, it's important to not cover the meat completely. Add just enough wine and broth to come up about 3/4 of the way to the top of the meat.