Rinse the bell peppers. Cut the tops off each pepper and remove and discard the seeds and membranes. Keep the pepper tops (lids) separate for later.
Place a large soup pot filled about 2/3 with water on the stove and bring to a boil. Add 2 Tbsp vinegar and 1 tsp salt (for the boiling water).
Add the peppers and their tops to the boiling water. Add more water if needed to cover the peppers. Cover the pot, boil 3 minutes, then turn off the heat and let the peppers sit in the hot water for 30–40 minutes to soften.
Before removing the peppers from the pot, use a heatproof measuring cup to remove and reserve 2 cups of the hot cooking water; set this aside. Remove the peppers and their tops to drain and cool enough to handle.
In a large skillet, heat 4 Tbsp olive oil over medium-high heat. Add 3 grated carrots, the chopped yellow onion, 2 cups chopped tomatoes, 1/4 cup chopped parsley, 1 tsp dried oregano, 1 tsp salt, and 1/2 tsp pepper. Sauté until the vegetables are soft. Add the 2 pressed garlic cloves and cook 1–2 minutes more. Remove the skillet from the heat.
In a large bowl, combine 2 lbs lean ground beef, 2 1/2 cups cooked white rice, 2 whole eggs, and 1 tsp salt. Add the sautéed vegetable mixture and mix until just combined.
Loosely fill each pepper with the beef-rice mixture (do not pack tightly). Place the stuffed peppers upright in a large, lidded pot or a Dutch oven. Set each pepper top (lid) back on its pepper.
Make the sauce: in a large skillet over medium heat, add the remaining 2 grated carrots and 2 cups chopped tomatoes (or 1 cup marinara). Stir until the tomatoes begin to break down, then add 1 Tbsp Mrs. Dash, 1/2 tsp salt, 1/2 tsp sugar, and 1 1/2 Tbsp sour cream. Stir to combine, then add the reserved 2 cups of pepper cooking water and stir to blend into a sauce.
Pour the sauce evenly over the stuffed peppers in the pot or Dutch oven so the peppers sit in the sauce. Cover the pot with its lid; also ensure each pepper has its individual top in place.
Cooking options: - Stovetop (large lidded soup pot): Bring the pot to a light boil over medium heat, then reduce to a simmer, cover, and cook 40 minutes, until the peppers are tender and the filling is cooked through. - Oven (Dutch oven): Preheat the oven to 450°F. Cover and bake the Dutch oven 20–25 minutes at 450°F, then reduce the oven temperature to 350°F and bake an additional 1 hour, until the peppers are tender and the filling is cooked through.
Remove from heat or oven and let the stuffed peppers sit briefly to cool slightly before serving.