An "atomic clock" comprises two major subsystems: a precision oscillator controlled by a feedback mechanism that's based on some quantum-mechanical phenomenon, and a digital counter/display subsystem that shows how many cycles of the oscillator have occurred so far.
The relative oscillator phase in multiple units can be measured; there are papers at NIST, and this article is worth a read for more general knowledge.
The counter mechanism can be reset at any time, and if you have two clocks, you can reset their counters from the same reference pulse. It's really entirely up to you how and when you reset your counters.
However, in some cases, it makes sense to just allow the counters to free-run, and instead sample and record their values at certain times, then compute the frequency and phase errors from those recordings. The computed values are then used to "correct" future readings from the clocks. This is the approach used with the atomic clocks used in GPS, for example.
It sounds as if you have a vague idea for a project, but that you haven't really given it a lot of thought or research yet. There are both professional and hobbyist websites devoted to the topic of precision chronometry — you might want to seek them out and do some reading to get an idea of what some of the practical issues are with respect to this topic.