1. Snap off and discard tough ends of asparagus; remove scales with a vegetable peeler, if desired.
2. Stir together 1 Tbsp. vegetable oil and garlic. Combine asparagus with vegetable oil mixture in a large bowl, tossing to coat. Sprinkle evenly with 1/8 tsp. salt.
3. Brush sweet onion slice evenly on both sides with 1 tsp. vegetable oil.
4. Grill onion slice and asparagus, without grill lid, over medium heat (300° to 350°). Grill onion slice 4 to 5 minutes on each side or until tender. Grill asparagus 2 to 3 minutes on each side or until tender.
5. Finely chop onion. Stir together onion, tomato, and next 3 ingredients. Serve salsa with warm asparagus.