Season pork with salt and pepper. Heat oil in a large heavy pot over medium. Cook pork, turning often until evenly browned, about 10 minutes. Transfer to a platter and pour off pan drippings.Add onion and garlic to pot and cook, stirring occasionally, until onion is starting to brown and caramelize, about 8 minutes. Add tomato paste and cook, stirring occasionally, until slightly darkened in color, 5 minutes.Add beef brith and cook, scraping up any browned bits, until reduced by about half, 5 minutes.Add tomatoes, then add thyme, rosemary. Add pork with any juices accumulated on the platter; season with salt and pepper.Bring liquid to a boil, then reduce heat and simmer, partially covered, until pork is falling-apart tender, the sauce is thickened (it will be thicker than a typical pasta sauce), and flavors have melded, about 40 minutes. In the last 5 minutes add the mushrooms. Using 2 forks, break up pork into pieces or shred it (your choice!); taste and season with salt and pepper.Pork can be cooked 3 days ahead. Let cool; cover and chill in the sauce.