Is the Torx head steel and the screw brass? If so, a magnet could potentially pull it out if it's not wedged too hard.
Also, can you still remove the screw with or without the chunk still in there without mangling the handle itself? If you can't get the bit of driver out but you can get the screw out with a screw extractor, your next best option is to track down and buy a replacement mounting screw for the handle. Depending on the make and model of the faucet, a replacement might be had with a quick trip to Ace/True Value/Home Depot/Lowes and either their plumbing or hardware departments (Ace and True Value will have the superior hardware selections)... otherwise, you may have to special order. Shouldn't be more than a couple-three bucks at screaming max, though.
Note for the future, though, and why you may have broken off a Torx head... it's not outside the realm of possibility that the screw is Torx, but usually handle screws on faucets are straight hex inset and require an Allen wrench. Don't substitute Torx for Allens, and pay for higher quality on both Torx and Allen tools for the smaller sizes, or you may find yourself repeating this incident. At minimum, get hardened chrome vanadium.