Brush garlic and onion slices with oil and place near the edge of a medium-hot grill to cook slowly. Grill for 5 to 10 minutes, turning garlic often. Leave on grill while cooking vegetables in step 2.
Cut eggplant into thick slices or in half lengthwise, depending on size. Brush eggplant, zucchini, yellow squash, and peppers lightly with oil and place in center of grill. Cook a few minutes per side, turning once, until vegetables are seared and brown.
Slice bread and brush with oil. Grill on both sides just until brown. Cut tomato in half and rub over one side of hot bread; discard tomato.
Arrange grilled vegetables, bread, onion, and garlic on top of arugula and serve immediately.