Preheat the oven to 300 degrees F. Bring the short ribs to room temperature. Meanwhile, pulse the shallots, 10 cloves garlic and the tomato paste in a food processor to make a paste. Cook the pancetta in a large ovenproof pot over medium-high heat until crisp, about 7 minutes. Transfer the pancetta and all but 2 tablespoons drippings to a bowl.
Add the shallot mixture to the pot and cook, stirring, until brown, about 6 minutes. Reduce the heat to medium and add 1/2 cup cognac, scraping up the browned bits. Sprinkle in the flour and cook, stirring, about 5 more minutes. Gradually stir in the wine until smooth. Add the porcini, thyme, bay leaves and 11/2 teaspoons salt.
Season the short ribs with salt and pepper, place in the pot and add water to cover. Trim a piece of parchment paper so that it rests on the surface of the meat. Cover the pot with the lid and transfer to the oven. Cook until the meat is fork-tender, about 3 hours.
Remove the meat from the sauce and transfer to a plate. Set the sauce aside, about 10 minutes, then skim off the fat. (If the sauce is thick, add a splash of water.) Discard the bay leaves, season the sauce with salt and pepper and add the remaining 2 tablespoons cognac. Return the short ribs to the sauce and keep warm over low heat.
Chop the remaining 1 clove garlic, then mash with 1/2 teaspoon salt using the flat side of a knife; add the parsley and chop. Heat the pancetta and the reserved drippings in a skillet over medium-high heat. Add the oyster and cremini mushrooms and cook until brown, about 10 minutes. Toss in the parsley mixture. Untie the short ribs, divide among plates and top with the sauce and mushrooms.