Yes I always did this.
FCC test houses are a bit more expensive but they can do both at the same time. They will ask you if you want to do this. It is much cheaper than doing them sepratly.
It is the FCC paper work that takes the time it could be up to six months and if you are not in the US you will need a filing agent.
As I said EU is just self certifying so you do the leg work with the test house backing up your claim.