Cartoccio means "paper" in Italian, which refers to the wrapping used to make packets for grilling. Here, foil packets preserve every drop of the delicious seafood juices for sopping up with crusty bread.
Light a grill. In a large bowl, toss all of the vegetables with olive oil and season with salt. Pull out the Broccolini and grill over moderately high heat until lightly charred, about 1 minute per side. Transfer to a large plate. Pull out the asparagus and carrots and grill for about 1 minute, until lightly charred; transfer to the plate. Grill the radishes, tomatoes and onion wedges, cut sides down, until lightly charred, about 2 minutes. Add to the other vegetables.
Tear off eight 16-by-18-inch pieces of heavy-duty foil. Layer the sheets in pairs. Divide the oysters, clams and mussels among the four pairs of foil and drizzle with olive oil. Arrange the vegetables over the shellfish and drizzle with more olive oil. Add a pinch of salt, a basil sprig and 1 tablespoon of water to each. Fold the foil tightly into neat rectangular packets.
Arrange the packets on the grill. Cover and cook over moderately high heat, rotating once or twice, until the packets are puffed and sizzling, about 25 minutes. Serve right away with bread.