Preheat oven to 300 degrees. Place Dutch oven over medium high heat and add 1 tablespoon of olive oil. Once hot, add chuck and sear until dark brown on all sides, about 3 minutes per side. Once seared on all sides, remove and set aside until later.
Add in onions, carrots, and celery and cook for 3 minutes or until soft. Add garlic and cook for 30 more seconds. Add 1 cup of beef broth and stir veggies to release pieces from the bottom of the pan. Add in balsamic vinegar, mustard, and brown sugar. Stir together to combine.
Place roast back into Dutch oven, moving some veggies around to make room for the roast. Add more broth to ensure half of the roast is covered with liquid. Add in the sprigs. Bring mixture to a light boil. Turn heat off and cover with a lid. Place in preheated oven and cook for 3.5-4.5 hours or until beef is tender enough to pull with a fork. Slice and serve or pull apart with forks. Add salt and pepper, if needed.