Preheat the oven to 400°F. Heat 2 tablespoons of vegetable oil in a large Dutch oven over medium-high heat. Season the short ribs on both sides with salt and pepper. Working in batches, add the short ribs to the pan and cook for about 4 minutes per side, until a golden brown crust forms. Transfer the short ribs to a plate and set aside.
Add the onion, carrots, celery, and garlic to the pan, season with salt and pepper, and cook for 4 to 5 minutes, until the vegetables are soft and caramelized. Add the flour and cook for about 1 minute, until well combined. Add the wine and cook for about 5 minutes, until the sauce thickens slightly. Place the short ribs back in the pan and add the beef broth, thyme, and enough water to cover the short ribs completely. Bring the mixture to a boil, then cover the pan and transfer to the oven. Cook for about 3 hours, until the meat is very tender.
Bring the water, milk, and 2 teaspoons of salt to a boil in a medium saucepan. Slowly whisk in the cornmeal, then reduce the heat to low. Cook for 15 to 20 minutes, stirring often, until the polenta thickens. Remove from the heat and stir in the Parmesan and cream cheese. Season with salt, to taste. Pour the polenta onto a large rimmed baking sheet and let cool.
Meanwhile, remove the beef short ribs from the pan and transfer to a cutting board. Using two forks, shred the meat into bite-size pieces. Return to the cooking liquid and keep warm.
Heat 1 tablespoon of vegetable oil in a medium skillet over medium heat. Form the polenta into six 2 to 3-inch round patties and dredge lightly in the flour. Working in batches, cook the polenta cakes for 4 to 5 minutes per side, adding more oil as needed, until golden brown. Transfer the polenta cakes to plates. Lift some of the short rib meat from the liquid and place over the cakes. Spoon some of the liquid over the top and garnish each plate with a sprig of fresh parsley.