Energy is also something I carefully consider for my post-MBA goals so I'll give my two cents on the matter.
I researched a lot about energy clubs and energy courses in each of the schools I applied to. I honestly came to the conclusion that, as Energy is not a major focus for MBA students, the intensity of exposure to the Energy field will depend heavily on where the school is located. Out of the 5 schools I researched, London Business School showed to have the strongest link to energy, partly due to the fact that London serves as the headquarters for many energy companies. On the other schools, the Energy Club did not stand out as a differential.
I guess both MIT and Yale will tell about their Energy Conference and the Houston trip that former students have done, but they'll not give you a "wow" factor.
On the other hand, as school boes, MIT and Yale are both Top-5 universities with a solid reputation all around the world. Their BS are in different positions, with Yale SOM climbing up the ranks, with more selective profiles each year where MIT Sloan is already one of the most selective, with a very strong reputation.
As you don't want to switch careers or even switch company, I would assume any of the schools will give you a great experience and both will accept you with arms wide open when you're back!
Congratulations on that great position to be in!