Preheat oven to 375F and preheat a grill to medium-high.
Place a medium pot over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add the onion and garlic to the pan, and cook until tender, 3-4 minutes. Add the tomatoes to the pan and break them up with a wooden spoon or potato masher. Add the cinnamon stick, salt, pepper and the basil to the pan, and simmer the sauce until thickened and aromatic, about 10 minutes.
While the sauce is simmering, brush the eggplant slices with the remaining EVOO and grill them until marked and tender, about 5 minutes per side.
Meanwhile, combine the ricotta, 1 cup Parmigiano Reggiano and egg yolks together in a medium bowl. Reserve.
When everything is ready, spread a thin layer of sauce over the bottom of a baking dish. Top with a layer of eggplant slices and spread half of the ricotta mixture over the eggplant. Top with another layer of eggplant and layer one third of the mozzarella over the eggplant. Top with some sauce and then more eggplant. Spread the remaining half of the ricotta mixture and top with another layer of eggplant. Top with another third of the mozzarella and sauce, and then with the remaining eggplant. Scatter the remaining mozzarella over top and sprinkle with the remaining grated cheese. Transfer the casserole to a baking sheet (in case it bubbles over) and bake until golden brown and bubbly, about 45 minutes. Garnish with basil and serve.