Preheat oven to 450°F. In a bowl, toss asparagus, artichokes, onion, 2 tbsp. oil, thyme and 1/2 tsp. each salt and pepper. Spread on a baking sheet. Place each garlic half, cut-side up, on a piece of foil. Drizzle each with 1/2 tsp. oil. Wrap tightly; place next to vegetables on sheet. Roast, stirring vegetables occasionally, until vegetables are golden and garlic is soft, 20 minutes. Unwrap garlic, squeeze out cloves and mash.
Bring a pot of water to a boil. Add 1 tsp. salt and pasta. Stir and cook until pasta is al dente, 9 minutes or according to package directions. Reserve 1/2 cup water; drain pasta.
In a large nonstick skillet, warm 1 tbsp. oil over medium heat. Add vegetables, garlic, vinegar, cooking water, pasta and 1/2 tsp. each salt and pepper. Simmer, stirring, until pasta is well coated. Top with Parmesan.