Prepare the lamb or beef - Trim any excess fat from the lamb or beef stew meat and season generously with salt and black pepper. In a Dutch oven or heavy-bottomed pot, heat the olive oil over medium-high heat. Sear the meat on all sides until browned, about 5 minutes per side. Transfer the meat to a plate and set aside.

Cook the onion and garlic - Add the chopped onion and minced garlic to the same pot and sauté for 5-7 minutes until softened and fragrant. Add the tomato paste and red pepper flakes and stir to combine. Cook for another 2-3 minutes until the mixture is thick and paste-like.

Slow-cook the meat - Return the meat to the pot and pour in the beef or chicken broth. Bring to a boil, then reduce the heat to low and let simmer for 2-3 hours until the meat is tender and falls apart easily with a fork. Check the pot occasionally and add more broth if needed.

Prepare the eggplant puree - While the meat is cooking, preheat the oven to 400°F. Slice the eggplants in half lengthwise and place them cut-side up on a baking sheet. Drizzle with olive oil and season with salt and black pepper. Roast in the oven for 25-30 minutes until tender and lightly browned. Remove from the oven and let cool for a few minutes. Scoop out the flesh with a spoon and transfer to a blender or food processor. Add the milk, butter, and all-purpose flour and puree until smooth and creamy.

Combine the meat and eggplant puree - Once the meat is tender, remove it from the heat and let it cool for a few minutes. Using two forks, shred the meat into bite-sized pieces. Add the eggplant puree to the pot and stir until well combined. Return the pot to the heat and cook for another 5-10 minutes until heated through. Serve the Hünkar Beğendi in bowls, garnished with fresh parsley or chopped tomatoes.