Prepare the seasoned flour. In a shallow bowl, combine 2 tablespoons all-purpose flour, 1 teaspoon salt, and ¼ teaspoon black pepper. Mix until evenly blended.
Coat the beef. Pat the beef chuck roast dry with paper towels. Coat all sides with the seasoned flour mixture, shaking off any excess.
Brown the pot roast. Melt the shortening in a large, heavy skillet with a tight-fitting lid or Dutch oven over medium-high heat. Add the beef and brown it on all sides.
Add the liquid and dill. Pour ¼ cup water and 1 tablespoon vinegar into the pan. Sprinkle 1 teaspoon dill weed over the roast.
Simmer the beef. Cover the pan tightly, reduce the heat to low, and gently simmer for approximately 3 hours, or until the beef becomes tender. Maintain a gentle simmer, adding a little water if necessary to prevent the pan from drying out.
Add the potatoes and carrots. Approximately 1 hour before the beef is expected to finish cooking, arrange the potatoes and carrots around the roast. Sprinkle with ½ teaspoon salt. Cover and continue simmering.
Add the zucchini. Approximately 20 minutes before the end of cooking, add the quartered zucchini. Sprinkle with the remaining ½ teaspoon salt, cover, and simmer until the vegetables are tender.
Transfer to a serving platter. Once the beef is fork-tender and the vegetables are cooked, transfer everything to a warm platter. Cover loosely to keep warm while preparing the gravy.
Reserve the drippings. Pour the pan drippings into a measuring cup, leaving the browned bits in the pan. Return 1 tablespoon of the drippings to the pan.
Start the gravy. Add 1 tablespoon flour to the reserved tablespoon of drippings. Whisk over low heat for approximately 2 minutes to form a smooth mixture.
Add the liquid. Add enough water to the drippings in the measuring cup to make 1 cup total liquid. Gradually whisk this mixture into the flour base.
Thicken the gravy. Bring the gravy to a boil while whisking constantly. Boil and stir for approximately 1 minute, or until thickened.
Finish with sour cream and dill. Season with salt and black pepper to taste. Reduce the heat to low and stir in 1 cup sour cream and 1 teaspoon dill weed. Heat gently until warmed through without allowing the gravy to boil vigorously.
Serve. Slice or portion the pot roast and serve alongside the potatoes, carrots, and zucchini. Spoon the warm sour cream dill gravy over the beef and vegetables.