Whisk all seasoning oil ingredients in a small bowl.
Brush eggplant slices with seasoning oil and place over medium-high heat on grill. Close lid and cook until tender, turning and brushing with remaining seasoning oil, about 5 to 10 minutes. Remove from grill. (You may also cook eggplant under the broiler, or saute them in a pan until tender, about 4 or 5 minutes per side.).
Place a slice of cheese on 1 eggplant slice, then top with another eggplant slice. Top this with 2 tomato slices or a piece of roasted red pepper, then put 2 to 4 basil leaves on top of that. Top with third eggplant slice, then another slice of cheese. Top with a fourth eggplant slice. Finish with a grind of pepper.
Repeat process to make 4 stacks. Place stacks on grill, close lid, and cook for about 2 minutes, turning once.
Serve immediately, or make into sandwiches on crusty rolls or bread. These sandwiches may be served warm or wrapped tightly and chilled several hours or overnight.