Put oven rack in middle position and preheat oven to 375°F.
Cut off and discard 1 end of baguette, then cut 12 (1/4-inch-thick) crosswise slices from baguette (reserve remainder for another use). Lightly brush 1 side of each slice with some oil (about 1 tablespoon total) and arrange, oiled sides up, on a baking sheet. Toast until golden, 8 to 10 minutes. While toasts are still warm, rub oiled sides with cut side of garlic clove half, then transfer to a rack to cool. Reduce oven temperature to 350°F.
Halve eggplant lengthwise and make shallow 1/2-inch-long incisions all over cut sides with tip of a paring knife. Arrange eggplant, cut sides up (without crowding), in a shallow baking dish and add unpeeled garlic clove. Sprinkle thyme, rosemary, oregano, sea salt, and pepper over eggplant, then drizzle eggplant and garlic with 2 tablespoons oil.
Bake until garlic is very tender, 30 to 35 minutes, then transfer garlic to a cutting board and continue to bake eggplant until very tender, 20 to 25 minutes more. When garlic is cool enough to handle, squeeze flesh from peel onto cutting board.
Transfer eggplant to cutting board and let stand until cool enough to handle, about 15 minutes. Scrape out flesh with a spoon onto cutting board, discarding peel. Finely chop eggplant and garlic together and transfer to a bowl. Add parsley and remaining tablespoon oil, then stir until combined well. Season with sea salt and pepper to taste.
Top toasts with eggplant mixture and sprinkle with cheese.