Heat oven to 480°F (250°C). Place peppers on foil and roast for 25-30 minutes (or until ready), turning occasionally as the peppers start to get charred.
While the peppers are roasting, sprinkle salt over eggplant rounds on both sides. Leave them to sweat out the bitterness while you make the beef.
In a deep skillet, heat about 4 teaspoons of oil over medium heat. Add onion, carrots, and parsley and simmer until the onion is translucent (about 5-6 minutes). Add meat and seasonings and simmer for another 10 minutes or so (add a pinch of water if necessary). Meanwhile, take your casserole pan and spread butter pieces on the bottom of it. When the meat (together with onion, carrot, and parsley) is done simmering, transfer it to the pan, spreading it evenly.
Take peppers out of the oven, and seal them in a plastic bag for a few minutes. Meanwhile, pat dry the eggplant with a paper towel. Add the remaining oil to the skillet and brown eggplant on all sides for about 7-10 minutes on medium, watching that it doesn't burn. Depending on the size of your skillet you may have to do it in batches. While the eggplant is frying, peel and deseed the peppers.
Layer finished eggplant over the meat in the casserole. Top everything with peppers and garnish with a few whole garlic cloves. Cover the dish with foil and pierce several holes using a toothpick or a fork.
Bake the casserole at 390-430°F (200-220°C) on the medium rack for 45 minutes. At this point, uncover and bake for an additional 5-10 minutes.
Serving: serve by itself or with a side. Side ideas include polenta, mashed potatoes, rice, pasta, and quinoa. Storing: store in a sealed container in the fridge for up to two days. Do not freeze! This roasted red pepper, eggplant, and beef casserole tastes even better the following day. Reheating: reheat using your favorite method. Add ¼ cup of water to the pan if necessary.