No, it doesn't support it. As I see HCT is a compatibility module
heavily dependent on LWP. IO::Lambda and LWP don't work together at all,
because LWP is blocking - which is both a necessary evil and pity, since
LWP has huge amount of code around it, and all that code should be
reimplemented for IO::Lambda, or anything else.
On a side note, I wonder why the author of HCT chose to overload
LWP::UserAgent::simple_request instead of providing a simple wrapper to
get(), like get_cached() to be used instead. That'd be more consistent,
I think. In the same fashion wrapping lambda's http_request() and
creating a http_request_cached() would be a similarly easy task.
Possibly the author could be interested in changing the module in such a
way so the caching logic would be detached from the actual get()
implementation, and so that it indeed could be used both in LWP and
IO::Lambda with minimal efforts?