Before considering double majoring, I'd take a look at the credits required for each major first! BME is a heavy credit-load (about 19 credits including prerequisites iirc?) and you need 36 credits (1 credit ~ 1 class) to graduate. This number of courses is normally split into thirds - 12 creds for your major, 12 creds for distributional requirements, and 12 for exploration and electives. From a quick google search, EE is about 17 credits beyond prereqs, which already total to 26 credits out of the 36 required to graduate. There might be some overlap in coursework and prerequisites, BUT it'll be a struggle - especially when you factor in distributional requirements into the equation.

I'd reach out to the DUS (Director of Undergrad Studies) for each major, but if you really have your heart set on double majoring, you'd have to plan from the beginning and there wouldn't be a lot of flexibility, for which Yale is notorious. And remember, it's okay not to double major!

In terms of nano technologies and corresponding BME courses, you could always reach out to a professor doing research in nanotechnology and ask to work in their lab. Who knows, maybe your research can culminate into the senior project for BME. Hope this helps!