Preheat the broiler. In a large, deep skillet, heat the EVOO over medium-high heat until the oil ripples. Add the lamb and cook, breaking up the meat with a potato masher or wooden spoon, until well browned; season with salt and pepper. Add the eggplant, onion, chile, garlic, cinnamon stick and oregano; partially cover and cook, stirring, until softened, 7 to 8 minutes. Stir in the tomato paste for 1 minute, then stir in 2 cups chicken stock. Lower the heat and simmer for 7 to 8 minutes more. Discard the cinnamon stick.
Meanwhile, in a medium saucepan, bring the remaining 2 cups chicken stock and the milk to a low boil. Whisk in the polenta and cook until thickened, 2 to 3 minutes; whisk in a little more water to thin the consistency, if desired. Stir in the butter, then the feta.
Spoon the polenta in an even layer onto the meat in the skillet and top with the parmigiano-reggiano. Broil for 1 minute to brown.