Place cooked pasta in a large bowl and toss with 1 tbsp of the oil.
Prepare a medium hot fire in the grill with a lightly oiled vegetable rack in place.
Combine the remaining 2 tbsp oil, garlic and rosemary in a large bowl. Add the onion, peppers and asparagus and toss to coat.
Grill the vegetables, tossing frequently, until lightly charred, about 5 minutes.
Add the grilled vegetables and radicchio to the pasta; toss with basil and lemon juice. Season with salt and pepper to taste. Serve at room temperature or chilled.
Please note that some ingredients and brands may not be available in every store.