Prepare a gas grill for direct-heat cooking over medium-high heat; see grilling procedure .
Pulse artichokes, mayonnaise, capers, and garlic in a food processor until coarsely chopped.
Trim off a thin slice from 2 opposite long sides of eggplant, then cut eggplant lengthwise into 4 (1/3-inch-thick) slices. Brush both sides of slices with 2 tablespoons oil (total) and season with 1/4 teaspoon each of salt and pepper.
Grill eggplant slices, covered, turning once and brushing grilled sides with 1 tablespoon oil (total), until golden-brown and tender, about 4 minutes, then transfer to a tray.
Brush both sides of bread with remaining 2 tablespoons oil and grill, covered, without turning, until grill marks appear, about 2 minutes.
Top each of 4 bread slices, grilled sides up, with cheese and an eggplant slice. Spread artichoke mixture on remaining 4 bread slices, grilled sides up, then assemble sandwiches.
Put sandwiches on grill and press down with a metal spatula, then grill, turning once, until heated through and grill marks appear, about 4 minutes total.
