Bridgespan is a nonprofit that does strategy consulting for nonprofits. Was started by some partners from Bain. I think they have a related company that offers nonprofit recruitment or something.
What you need to find out is how Monitor Institute hires. Is it seperate from the general consulting firm? If yes it's likely very limited and self directed. If it is through campus with the consulting firm then you need to find out how much work they do in this area relative to other work. Also, you need to decide if you are okay with doing some work in this area, but working on other projects as well. I know Monitor doesn't specialize by industry so you likely wouldn't be able to work exclusively on this type work. Might be able to increase the percentage after you get experience, build relationships etc
Just something to think about....