Heat olive oil in a skillet over medium heat; cook and stir onion and carrot until onion is translucent, about 5 minutes. Pour tomatoes into skillet and sprinkle sugar, salt, and black pepper into mixture. Bring to a simmer, turn heat to low, and simmer while you complete remaining steps. Stir occasionally.
Heat vegetable oil in a separate skillet over high heat and fry eggplant slices in the hot oil until lightly browned, about 2 minutes per side. Drain eggplant slices on paper towels.
Fill a large pot with lightly salted water and bring to a boil. Stir in tortiglioni pasta and return to a boil. Cook pasta uncovered, stirring occasionally, until cooked through, but still firm to the bite, about 11 minutes; drain.
Mash tomato sauce with a fork until chunky and transfer to a large bowl. Stir in eggplant slices, Kalamata olives, Grana Padano cheese, mayonnaise, and pasta; stir lightly to incorporate.