Heat grill or grill pan to medium. Brush both sides of the eggplant with 1 tablespoon of the olive oil and sprinkle with the salt.
Grill until tender and slightly charred, 4 to 5 minutes per side.
Spread 2 slices of bread with the tomato paste and spread the remaining 2 slices with the goat cheese. Form 2 sandwiches with the bread and eggplant.
Brush both sides of the sandwiches with 1 tablespoon of the oil, then press on 1 tablespoon of the Parmesan. (The oil helps the cheese stick.)
Heat the remaining tablespoon of oil in a large nonstick skillet over medium-low heat.
Add the sandwiches and cook until the Parmesan has browned, 4 to 5 minutes per side.