Preheat an outdoor grill or a grill pan to medium-high. Brush both sides of the eggplants with EVOO (about 1/4 cup total) and season with salt and pepper. Grill, turning occasionally, until tender, 10 to 12 minutes. Transfer to a plate and set aside. Grill the capicola slices for 1 minute on each side.
While the eggplant cooks, in a medium saucepan, heat the remaining EVOO on the grill or over medium-high heat on the stove. Add the onion and garlic and cook until softened, 5 minutes. Add the vinegar and stir for 30 seconds. Add the tomatoes and cinnamon; season lightly with salt and heavily with pepper and simmer for 10 minutes. Stir in the basil.
Pour a little of the sauce into a shallow, flameproof baking dish and spread to cover the bottom. Make 4 stacks of 2 repeating layers of eggplant, capicola and cheese. Top each stack with the remaining eggplant slices, some sauce and the remaining slices of cheese.
Set the baking dish on the grill and cover the grill (or place under the broiler), and cook until the cheese melts. Carefully transfer the stacks to plates, spooning extra sauce on the plate.