Mix 1 tablespoon flour, salt, and pepper together in a shallow bowl. Coat entire beef roast with flour mixture.
Heat olive oil in a large pot over medium heat; cook beef roast in the hot oil until browned on all sides, about 5 minutes. Add 1/4 cup water, vinegar, and 1 teaspoon dill to the pot. Cover pot and simmer until pot roast is tender, about 2 hours.
Arrange potatoes and carrots around roast; cover pot and cook until potatoes are tender, about 40 minutes. Add zucchini to roast mixture; cook until zucchini is tender, about 20 minutes.
Transfer roast and vegetables to a warm serving platter; cover loosely with aluminum foil. Pour the remaining drippings into a bowl.
Whisk 1 tablespoon drippings and 1 tablespoon flour together in the same pot; cook and stir over low heat until smooth and bubbly, 1 to 2 minutes. Remove pot from heat. Pour remaining drippings into a measuring cup, adding enough water for total liquid to equal 1 cup.
Return pot to stove and stir drippings-water mixture into flour mixture; bring to a boil and cook gravy for 1 minute. Remove pot from heat and season gravy with salt and pepper. Stir sour cream and 1 teaspoon dill into gravy until smooth. Heat gravy over low heat until warmed, 2 to 3 minutes; serve alongside roast and vegetables.