Just try and sit on your hands and let that first call come from him. There is a big difference between calling a man randomly and calling him because he requested that you do so.
You could tell him to hold on and you will write your number down for him so he can call you when it is convenient or you could just take the number and call him later.
Either way this is a time when calling a man is acceptable.
If he offers you his then just tell him you will get it from him when he calls you. There will be no doubt that it is understood that you expect him to call you.
If he does not call then do not fret, it wasnt meant to be.
Does this sound a lot like what your grandmother told you?
This is something that will never change, no matter how much society changes.
In fact there are entire groups of people that do not agree with this but since the beginning of time men have been much more interested in something that is a bit of a challenge than something that comes too easily.
Think about what the man is instinctively meant for.
Calling After the First Date This is a really tough one.