Could you smoke a sauce?
What you're describing isn't all that different from how they make various products like Liquid Smoke (make smoke along with steam, then condense that steam).  You will need to make sure that some actual condensation occurs (for example, by having a lid for the smokey vapor to condense onto).

However, it may be simpler to add a liquid smoke-type product directly; at least the main brands are actually smoke distillates (and not artificial flavorings or similar) and thus should be effectively identical to this process, but quicker.