Combine the marinade ingredients, from the smashed garlic cloves through the thyme, in a large bowl.
Place the short ribs in a large, sealable plastic bag. Carefully pour in the marinade, and seal the bag. Gently toss the marinade so that the ribs are well coated
Put the short ribs into the refrigerator and marinate for at least 8 hours.
When you are ready to start cooking, remove the short ribs from the fridge. Place a strainer in a large plastic bowl in the sink. Pour the short ribs and marinade into the strainer. Reserve the extra marinade in the bowl, as you will use 2 cups of it in the slow cooker.
Take the short ribs out of the strainer, brushing off any of the solid marinade ingredients that may be sticking to them. Pat the ribs dry and set aside on a plate. Pick out 3 of the garlic cloves from the strainer and reserve for use.
Heat 1 tablespoon of the olive oil in a large skillet. Add the onions and carrots, cook for about 5 minutes, stirring occasionally.
Clear a space in the skillet to add the short ribs. Brown the short ribs on all sides by turning occasionally. This will take 5 to 10 minutes. You may need to brown the short ribs in two batches depending on the size of your skillet.
Move the short ribs with the onions and carrots to the slow cooker, scraping all the bits of vegetable and beef into the slow cooker.
Pour the 2 cups reserved marinade and the beef broth over the short ribs. Add the 3 reserved garlic cloves and a sprig of rosemary.
Cover the slow cooker, and cook on low heat for 10 hours.
When the short ribs have finished slow cooking, move them to a platter to cool. Reserve the cooking liquid.
As the short ribs are cooling, heat 1 tablespoon olive oil in a large pot. Add the onion, cook on medium heat until it begins to soften, about 5 minutes.
Stir the mushrooms into the pot, along with 1/2 tsp dried thyme and salt to taste. Cook until the mushrooms begin to sweat about 5 minutes.
Add the 1/4 cup red wine, and cook for another 5 minutes until the wine has mostly been absorbed by the mushrooms.
As the onions and mushrooms cook, remove any beef from the bones, and set the bones aside. Shred the short rib meat with a fork, discarding any large pieces of fat. Add the meat to the mushrooms, and stir in the 1/2 cup cooking liquid (or more liquid if desired). Simmer until the meat is heated and the liquid has been reduced by half. Add salt and pepper to taste.
Serve the short ribs and mushrooms over mashed potatoes or noodles. Enjoy!