Step 1Heat oven to 325 degrees. Season the short ribs lightly with salt and pepper. Heat a Dutch oven over high heat until hot. Add vegetable oil and sear the short ribs on all sides. Remove the short ribs and set aside on a plate.
Step 2Add the onion, carrot, celery and garlic to the Dutch oven and cook, stirring frequently, until the vegetables are caramelized, 10 to 12 minutes. Reduce the heat to medium and skim any excess fat from the top.
Step 3Stir in the tomato paste, then stir in the flour. Add dry red wine, thyme, bay leaves, rosemary, the seared short ribs and enough beef broth to cover the meat (add water if necessary), then bring to a boil, skimming any fat off the top of the liquid as it cooks.
Step 4Cover the Dutch oven and place in the oven until the meat is tender, about 2 1/2 hours, checking every 20 to 30 minutes to skim excess fat off the top. Add additional broth or water as necessary to keep the meat submerged in cooking liquid.
Step 5Meanwhile, heat a saute pan over high heat and add the butter. When the butter is melted, add the mushrooms. Cook until the mushrooms are caramelized, stirring frequently, and season lightly with salt and pepper. Remove from heat.
Step 6When the meat is fully cooked, carefully remove it and set aside on top of the mushrooms.
Step 7On the stove top, bring the braising liquid back to a boil, removing excess fat as necessary. Reduce the braising liquid to about 4 cups, then strain. Place the liquid back into the Dutch oven and add the cream. Season to taste and simmer over low heat until the sauce thickens to a creamy consistency, about 15 minutes. Gently stir in the meat and mushrooms before serving.