For the basil oil: combine the olive oil and basil leaves in a food processor and season with salt and pepper. Process until pureed (there will be some sediment), about 5 minutes. Strain into a squeeze bottle. May be refrigerated for 1 day; use at room temperature. Makes about 1 1/4 cup.
For the sandwiches: combine the olive oil, balsamic vinegar, and garlic and divide between two large bowls. Add the eggplant to one bowl and the tomatoes to the other and let marinade 1 hour, refrigerated.
Preheat a gas or charcoal grill to high.
Remove the eggplant and tomatoes from the marinade, shaking off any excess, and season with salt and pepper on both sides. (Discard the used marinade). Grill the eggplant until golden brown and cooked through, about 3 minutes on each side; grill the tomatoes just until marked.
Place 8 slices of eggplant on a work surface and top each with a slice of tomato. Place a slice of goat cheese over the tomato and sprinkle with the basil (some basil will be left over). Cover with the remaining eggplant slices. Place the sandwiches on the grill just until the cheese has softened, about 1 minute, turning once (DH says he won't turn the sandwiches next time as it made them a bit messy).
Arrange the eggplant sandwiches on a platter and drizzle with the basil oil.