Preheat an oven to 375F.
Tear the pita bread into bite-size pieces and arrange on a sheet tray. Toast in the oven until golden brown, 8-10 minutes.
Place a large skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Once hot, add the ground lamb and break it up with the back of a wooden spoon as it browns, 6-8 minutes. Add the eggplant, garlic, onion, oregano, cinnamon and paprika and season with some salt and freshly ground black pepper. Put a lid or some foil over the skillet and let everything cook for 5 minutes. Remove the lid, add 1 cup of beef stock to and continue to cook for another 4-5 minutes.
While the chili is cooking, make a bchamel sauce: Melt the butter in a medium skillet over medium-high heat. Add the flour and cook for about 2 minutes. Whisk the milk into the roux and bring to a boil. Reduce the heat to medium and simmer until the sauce has thickened, stirring frequently. Season with salt, freshly ground black pepper and a pinch of nutmeg.
Divide the toasted pitas between four serving bowls. Ladle the bchamel sauce over the chips and top with some chili. Garnish each bowl with some feta, chopped pepperoncini, olives, tomatoes and parsley.