Season the short ribs with the salt and pepper. Heat a large pot over medium-high heat and add the olive oil. Once it’s hot, add the short ribs and sear them until deeply golden, about 4 to 5 minutes. Add in 2/3 cup red wine and let cook for 1 to 2 minutes, the reduce the heat to low and cover. Cook the short ribs for 2 hours, checking on them occasionally. If the liquid cooks off, add more in the form of wine, beef stock or water just so there is a thin layer on the bottom of the pot.
After 2 hours, remove the ribs from the pot and place them in a bowl. There should be a bit of liquid left in the pot (if there seems to be a lot, drain some). Add the mushrooms and cook, stirring until softened, about 5 minutes. Add in the onions and garlic, stirring to toss. Cook for 5 minutes. Add the remaining wine and cook for 2 minutes more. Remove the beef from the ribs, discarding any large chunks of fat or gristle. Add the beef, beef stock and potatoes to the pot. Bring the mixture to a boil then cover and reduce the heat to low. Cook for 20 minutes, or until the potatoes just soften.
Place the cold water in a shaker bottle. Add the flour to the water and cover the bottle (or jar) and shake for 30 seconds to make a slurry. Add the carrots and the slurry to the stew and stir well. Increase the heat a bit until the stew is simmering. Let it cook for 30 more minutes, stirring occasionally, until the carrots are softened. Taste and season additionally with salt and pepper if needed – this will depend on the sodium level of your stock.
Serve with freshly chopped parsley on top.