Is there a method of sword fighting where one uses two handed techniques with a basket-hilted sword? I've heard that this happened, just not where, when, and what the technique was called.
There is no space in a basket hilt to use two hands, so the second hand would be on the blade. It became more common with all swords as armour improved.
Historically speaking, there wouldn't be much overlap between the us of the half-swording technique and the advent of the basket-hilted sword. Half-swording was typically used with a longsword by heavily armored combatants. The basket-hilt arose once technology had advanced to the point that heavy armor began to fall out of favor. The basket-hilt itself being a design to protect the unguantleted hand. Against the lightly armored soldiers of the day a basket-hilted blade would not require a technique like half-swording, but there may have been the odd exception (perhaps some twit who refused to retire his Maximillian-style fluted plate).