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.
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).
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.
- There is no way to wield a proper basket hilt sword two-handed.
This is because the enclosure of the basket shape is specifically designed to protect the sword hand by covering as much of the area around the and as possible.
When wielding two-handed, such as with a longsword, the forearms come in from different angles. The most coverage you could get would be 180 degrees to the front, if that, and it would still limit a range of potential techniques.
The only thing I've seen proposed is the two-handed Swiss saber,but I wouldn't categorize that limited hand guard as a basket hilt.
The fact that some Swiss sabers have such guards reinforces the vulnerability of the hands in combat with long blades, where the hands/wrists are usually the most accessible target at range.
The introduction of some form of hand protection may relate to transition from steel gauntlets to leather gloves around that general time period, but the design doesn't seem to have caught on, and the use of the Swiss saber was probably specialized.
The lack of steel gauntlets in the basket-hilt era would make half-swording too risky, and there's not much reason to do it with a single-handed sword.