Whenever a skill, ability, or item grants a turn within a turn, or an engage within an engage, or a (non-interrupt) class ability within an engage - pause the current turn/engage, and resolve the thing "within" as if it is separate. The damage dealt is dealt in the turn/engage/ability within, it is not held to be later dealt with the damage of the encompassing turn/engage. Once you've finished the turn/engage/ability within, unpause the encompassing turn/engage and continue on.
They are two separate turns/engages, even if one is happening within another.