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.
Cook's Note: 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.