Heat the oil in a high-sided large skillet or Dutch oven over high heat. Add 2 pounds of cherry tomatoes and cover tightly. (They will spatter.) Cook, shaking the pan frequently for 8 minutes. The tomatoes need to brown and break down. Reduce the heat to medium, uncover the pan, and pop the tomatoes with a wooden spoon. Stir in the oregano, sugar, salt, and freshly ground black pepper. Cook for one more minute; set aside.

Combine the ricotta, pesto, and salt in a bowl; set aside.

Cook the lasagna noodles according to package directions.

Preheat the oven to 425ºF. Lightly oil a 9x13-inch baking dish. Spread half of the cherry tomato sauce on the bottom of the dish.

Lay all of the lasagna noodles out on a work surface. Divide the ricotta pesto mixture equally between the noodles and spread the mixture out on each noodle. Place two slices of sliced tomato on each and carefully roll the noodles up—season with salt and pepper. Place the rolls, seam down, and side by side, in the baking dish. Cover the rolls with the remaining sauce and arrange the halved cherry tomatoes on top.

Cover the dish tightly with foil and bake for 30 minutes.

Remove the pan from the oven and remove the foil, sprinkle evenly with the Parmesan, and return to the oven to bake for an additional 10 minutes. Cool the rolls for 15 minutes, uncovered; garnish with fresh basil leaves and serve.