Trim tops from each bell pepper and remove seeds and membranes; keep the tops (lids) if desired.
Fill a large pot about two-thirds full with water and bring to a boil. Add 2 tablespoons vinegar and 1 teaspoon salt.
Place the peppers and their tops into the boiling water so they are mostly covered. Cover the pot, boil 3 minutes, then turn off the heat and let peppers sit in the hot water for about 30–40 minutes. Reserve 2 cups of the cooking water before draining.
In a large skillet, heat 4 tablespoons olive oil over medium-high heat. Add 3 grated carrots, the chopped onion, 2 cups chopped tomatoes, parsley, 1 teaspoon dried oregano, 1 teaspoon salt, and 1/2 teaspoon pepper. Sauté until the vegetables are soft, about 5–8 minutes. Add the pressed garlic and cook 1–2 more minutes, then remove from heat.
In a large bowl, combine 2 pounds ground beef, 2 1/2 cups cooked white rice, 2 beaten eggs, and 1 teaspoon salt. Add the sautéed vegetable mixture and mix gently until evenly combined.
Loosely fill each pepper with the beef and rice mixture; do not pack tightly. Arrange the stuffed peppers upright in a dutch oven or large lidded pot.
Prepare the sauce: in a skillet, sauté the remaining 2 grated carrots with 2 cups chopped tomatoes (or 1 cup marinara) over medium heat. Stir in 1 tablespoon Mrs. Dash (or seasoning), 1/2 teaspoon salt, 1/2 teaspoon sugar, and 1 1/2 tablespoons sour cream. Add the 2 cups reserved pepper cooking water and stir until combined.
Pour the sauce evenly over and around the stuffed peppers, then place the pepper tops back on if using.
If cooking on the stovetop in a lidded pot: bring the liquid to a gentle boil, then reduce heat, cover, and simmer for 40 minutes.
If using a dutch oven (recommended): cover and bake at 450°F (232°C) for 20–25 minutes, then reduce oven temperature to 350°F (177°C) and bake an additional 60 minutes, until peppers are tender and filling is cooked through.
