Keeping old job AND taking new opportunity?

I work in an extremely niche segment of the Energy sector. I currently work for one company, and was head hunted for a director position with a competitor. I genuinely enjoy the team at my current company and my departure will put them in a tight spot with our current schedule and project load. However, I cannot turn down the new position with the pay increase and ability to build my own department.

Has anyone offered to stay on at a previous employer as a consultant or a sort of professional and transparent moonlighting until a replacement is found?

I cannot double employ, as I regularly interact with third party engineering firms that both companies partner with.