In a large bowl, whisk together oils, vinegar, parsley and garlic.
Slice eggplant and tomatoes into rounds. Cut cheese into pieces no bigger than vegetable rounds.
Make parmigianas: On a cutting board or other flat surface, place a slice of eggplant, cover with a slice of cheese, a tomato slices, another layer of cheese, then a slice of eggplant. Pierce through with a toothpick to prevent it from coming apart. Place each parmigiana in a shallow dish, and drizzle with vinaigrette. Cover and refrigerate 1 hour.
Preheat grill to medium-high and lightly oil grill rack. Grill parmiginanas on top rack for 5 minutes per side or until eggplant is tender and golden and cheese is melted.