Preheat the oven to 400 degrees F.
Place the cherry tomatoes on a large baking sheet along with the thyme, oregano and garlic. Dress with 2 tablespoons of the EVOO, season with salt and pepper and toss to coat. Roast until the tomatoes burst, 35 to 45 minutes
Brush the eggplant with the remaining 2 tablespoons EVOO and sprinkle with salt and pepper. Place cut-side down on a baking sheet and roast alongside the tomatoes until soft, 30 to 40 minutes.
Let the tomatoes cool slightly, and then puree them with the basil in a food processor until smooth. Remove the skins from the garlic and puree the sauce with the roasted cloves before transferring the sauce to a bowl.
When the eggplant has cooled, spoon out the flesh and add it into the cleaned bowl of the food processor along with the ricotta, Parmigiano-Reggiano, egg yolks and some salt and pepper. Puree the mixture until smooth and season with more salt and pepper.
Ladle one-third of the tomato sauce into the bottom of a 9-by-13-inch casserole dish. Layer pasta sheets on top, followed by one-half of the eggplant, more pasta sheets, tomato sauce and 1/2 of the mozzarella. Repeat with another layer of pasta sheets, followed by the remaining eggplant, more pasta sheets and the remaining tomato sauce. Top the lasagna with the remaining mozzarella. Bake 1 hour. Let cool 20 minutes before serving.
Cool completely before storing in the refrigerator or freezer for a make-ahead meal. Bring to room temperature before reheating, covered, at 325 degrees F, 20 to 30 minutes. Uncover and bake an additional 10 to 15 minutes.
Recipe courtesy of Rachael Ray