Prep the mushrooms and veg - Cut the carrots ins a fine dice. Peel the onion and cut it into fat slices. Destem the mushrooms, then tear or slice them into bite-sized pieces.
Preheat the oven and brown the chuck - Preheat the oven to 275℉. Cut the chuck into large chunks, and salt and pepper all over. Brown the beef on all sides in the pot. Set the beef aside on a large plate.
Sauté the mushrooms - Add the mushrooms to the pot and sauté until the mushrooms have softened, scraping the fond off the bottom of the pot as the mushrooms cook. Once the mushrooms are softened, scoop them out of the pot and move them to the plate with the beef.
Build the braise - Add the carrots to the pot and cook over medium high heat until they begin to soften. Stir in the tomato paste, and cook for a minute or so until the mixture begins to darken, then stir in the mustard. Push the carrot and tomato paste mixture to the outer edges of the pan, and lay the thick slices of onion on the bottom of the pan, in the center. Pour the beef chunks and mushrooms, and any accumulated juices, on top of the onions. Pour in the wine and the balsamic, then add the broth, pouring it only to the shoulders of the beef. Reserve any extra broth in case you want to add volume to the sauce you make at the end.
Braise - Cover the pot and place it in the center of the oven. Braise for 3-4 hours, or until the chuck roast is fall-apart tender. Submerge the rosemary sprigs into the braising liquid near the end of the cooking time. Don't peak until the 2.5-hour mark - just let it stew and simmer until then. Chop enough parsley, stems and leaves, to measure 1/2 cup.
Finish - Pull the pot out of the oven and place it on the stovetop. Using a slotted spoon, scoop the beef and the mushrooms out of the sauce and set aside. Strain the sauce (discarding the veg and the rosemary) and de-fat it using a fat separator, and return it to the pot. Whisk the water and cornstarch together, then add to the sauce. Cook on medium heat, stirring often, until the sauce begins to thicken. Remove from the heat and return the beef and mushrooms back to the pot.
Serve - Serve on a base of mashed potatoes or polenta. Garnish with chopped parsley and serve.