Place sliced eggplant into a large pot. Cover with salt and add enough water to cover. Place a plate on top of the rounds to keep them submerged; allow to sit for 10 minutes.

While eggplant soaks, whisk egg in a shallow dish. Combine bread crumbs, oregano, black pepper, and cayenne pepper in another shallow dish.

Heat oil in a deep skillet over medium-high heat.

Drain eggplant and discard water. Rinse eggplant and squeeze out as much water as possible. Pat slices dry with paper towels. Dredge eggplant slices in egg and then bread crumb mixture and place on a plate. Dredge halloumi cheese sticks in egg and then bread crumb mixture and place on a plate.

Fry breaded eggplant slices in the hot oil until browned, about 2 minutes. Flip and fry on the other side, about 2 minutes more. Drain on clean paper towels. Repeat with halloumi cheese sticks.

Meanwhile bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Toss cooked pasta with extra-virgin olive oil.

Pour spaghetti sauce into a small saucepan and heat over medium heat until hot, about 5 minutes.

Divide cooked spaghetti onto 4 plates. Evenly distribute eggplant and halloumi cheese. Top with equal amounts of spaghetti sauce and Parmigiano-Reggiano cheese.