Prepare the grill for indirect cooking over medium heat (350° to 450°F).
Remove the loose, papery outer skin from a head of garlic and cut off the top to expose the cloves. Drizzle the oil over the cloves. Replace the top and place on a large square of aluminum foil. Fold up the sides and seal to make a packet, leaving a little room for the expansion of steam.
Brush the cooking grates clean. Grill the garlic packet over indirect medium heat, with the lid closed, until the cloves are soft, 45 minutes to 1 hour. Remove the garlic packet from the grill and allow to cool. Unwrap the packet and squeeze out the garlic cloves into a food processor. Add the remaining paste ingredients and process until smooth.
Cut the eggplant crosswise into slices about ½-inch thick. Cut the bell peppers lengthwise in half. Remove the stems, core, and seeds and flatten the peppers with the palm of your hand. Lightly brush the eggplant and bell peppers with oil and season with salt and pepper. Grill over direct heat, with the lid closed, until tender but not limp, turning once. The eggplant will take 8 to 10 minutes and the bell peppers will take 6 to 8 minutes. The bell pepper skins should be seared with grill marks, but not blackened and peeling. Remove from the grill as they are done.
Related Grill Skills
Just before serving, toast the rolls, cut side down, over direct heat, about 30 seconds. Brush the inside of the rolls with the garlic paste and layer with the grilled vegetables, lettuce, and goat cheese. Serve warm or at room temperature.