Preheat the oven to 375 degrees F. Preheat a grill to medium-high.

Place a medium size pot over medium-high heat. Add 2 Tbsp. oil. Add the onion and garlic to the pan and cook until tender, about 3-4 minutes.

Add the tomatoes to the pan and break them up with a wooden spoon. Stir in the salt, pepper and the basil to the pan and simmer 10 minutes.

While the sauce is simmering, brush the eggplant slices with the remaining oil. Place eggplant slices on grill and cook until they're marked from the grill and tender, about 5 minutes per side.

Meanwhile, combine the ricotta, 3/4 cup Parmigiano cheese and egg yolks in a medium bowl. Reserve.

Spread a thin layer of tomato sauce in 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.

Place the remaining mozzarella over the top and sprinkle with the remaining 1/4 c Parmigiano.

Place the casserole dish on top of a baking sheet (in case it bubbles over) and bake until golden brown and bubbly, about 45 minutes. Garnish with basil and serve.