Stir together oil, garlic, thyme sprigs, and crushed red pepper in a small saucepan. Cook over low heat, undisturbed, until garlic is softened and golden brown in spots, about 15 minutes. (To maintain a gentle simmer, position saucepan over stovetop burner so that heat source only partially touches bottom of saucepan.) Remove from heat; let cool 15 minutes. Remove and discard thyme sprigs.

Preheat oven to 450°F with racks in middle and lower third positions. Brush eggplant rounds, bell pepper halves, and mushroom caps all over with ¾ cup garlic oil; season vegetables on both sides with salt and black pepper. Divide mixture evenly among 2 wire racks set inside 2 rimmed baking sheets, and arrange in an even layer. Roast until vegetables are tender and browned in spots, about 25 minutes, turning vegetables and rotating baking sheets between top and bottom racks halfway through cook time.

While vegetables roast, make olive relish: Mash garlic cloves in remaining garlic oil in a small bowl until mostly smooth. Stir in olives, giardiniera, giardiniera brine, and capers.

Spread about 2 tablespoons olive relish over each bottom roll half; top with 1 mushroom cap, 2 eggplant rounds, 2 bell pepper halves, and 2 cheese slices. Spread each with about 3 tablespoons olive relish, and top sandwiches, pressing down. (Reserve remaining relish for another use.) Serve immediately. Or, for the full New Orleans experience, wrap sandwiches individually in parchment paper followed by aluminum foil. Place on a baking sheet, and top with another baking sheet topped with a cast-iron skillet, a brick, or other heavy items such as large unopened cans. Let stand 1 hour at room temperature. Unwrap sandwiches, and serve.