How to stop a lye + oil + water reaction in aluminium container?
Slowly add vinegar. DO NOT TURN THE FLAME BACK ON.

Most likely, the alkali concentration is high enough that you've dissolved away the passivating oxide on the aluminum, and the exposed aluminum metal is reacting with water to form $\ce{Al(OH)4-}$ and hydrogen gas.  Adding vinegar should bring the $\mathrm{pH}$ back down into a range where the aluminum can passivate again, stopping the reaction.

The batch of soap is probably already ruined, so I wouldn't worry about that.

In the future, I would recommend using a cast iron or mild steel (not stainless) pan for this procedure.  Plain iron is remarkable for its resistance to alkali, in my experience.