Trim and cut the Lamb Shoulder into bite-sized cubes, then season generously with Salt and Black Pepper.
Heat a few tablespoons of Olive Oil in a heavy-based pot over medium-high heat and brown the lamb in batches until well colored; transfer the browned pieces to a plate.
Reduce the heat to medium and add the chopped Onion to the same pot; sauté until translucent, then add sliced Green Turkish Pepper and cook until softened.
Stir in minced Garlic and cook for 30–60 seconds until fragrant, then add chopped Tomato and a spoonful of Tomato Paste; fry everything together a couple of minutes to deepen the flavors.
Return the browned lamb to the pot, tuck in a Bay Leaf, pour in enough Water to barely cover the meat, bring to a gentle simmer, cover and cook 1–1½ hours (or until fork-tender), skimming any foam and adjusting Salt and Black Pepper to taste.
While the lamb braises, char whole Eggplant under a hot broiler or over a grill until the skin is blackened and the flesh is very soft (about 25–40 minutes); let cool, peel off the skin, drain any excess liquid and mash or purée the flesh coarsely.
Make a béchamel base: melt Butter in a saucepan over medium heat, whisk in Flour and cook 1–2 minutes without browning, then slowly whisk in warmed Milk until the sauce is thick and smooth. Season lightly with Salt and Black Pepper.
Stir the roasted eggplant into the béchamel, add grated Aged Kaşar Cheese and a splash of Lemon Juice, and mix until the cheese melts and the purée is creamy and well combined; adjust lemon and seasoning to brighten the flavor.
To serve, spread a generous layer of the eggplant purée on warmed plates or a serving platter and spoon the braised lamb and its sauce over the top. Optionally finish with a drizzle of Olive Oil or extra grated cheese and serve hot.
