Preheat oven to 350 degrees. Trim the chuck roast, removing large strands of visible fat and cutting the roast into 6-8 pieces. In a 5-7 qt. Dutch oven, heat oil over medium-high heat. Add the meat in a single layer and brown for 3-4 minutes on each side (meat may need to be browned in 2 batches). Remove the meat to a plate.
Add remaining 2 T. oil, onions, mushrooms and sun-dried tomatoes to the Dutch oven. Saute until the onions are tender. Add the flour and stir. Add the beef stock and deglaze the pan by scraping up any brown bits that have formed on the bottom of the pan (they have intense flavor). Add the balsamic vinegar, red wine, garlic, smoked paprika and salt and pepper. Stir until ingredients are combined, then return the chuck roast pieces and juices to the dutch oven.
Lay a piece of aluminum foil on top and carefully press down until it touches the entire surface of the pot roast. Tightly seal the edges of the foil, crimping them over the top of the dutch oven. Put lid on top. Place in the lower half of the oven and allow to braise for 2 hours.
Hot meal on a cold day or something hot is really soothing over cold winter evenings. Here is a recipe by Chef Ginny to make Balsamic Glazed Pot Roast with chuck roast, some vegetables and seasonings. Meaty affair to kill the cold!