Preheat the oven to 325 degrees F (165 degrees C).

Heat vegetable oil in a Dutch oven over medium heat. Cook roast in hot oil until browned on all sides, 2 to 3 minutes per side.

Mix together brown sugar, salt, mustard, and black pepper in a small bowl; sprinkle evenly over roast. Arrange potatoes, carrots, and onion in the Dutch oven around roast. Pour vinegar and water over vegetables.

Cover and cook in the preheated oven until roast is hot in the center and just turning from pink to grey, about 4 hours. An instant-read thermometer inserted into the center should read 150 degrees F (65 degrees C).