Well, it sort of depends on the atty or carto and what type it is. In any case, if it's an atty blow out the "primer" (whatever that is) before you use it and consider (but it's not essential) cleaning it out.
You don't really break them in but they do need time to wick the first ever
juice you put in there.
The cartos I use I'll fill them up for the first time and give it 10 - 30 minutes to absorb it all then it's good to go, no breaking in required.
For my atties, my tank atties need broken in to a small degree but it's more along the lines of getting
juice inside the things. Like you suggested I'll just vape them without inhaling the vapour until I can taste it and it's producing the right amount but it's more because I use all VG and it takes a long time to wick with tanks and less about breaking it in. Atties with carts you can simply direct drip to get an instant wicking so again it doesn't really need broken in.
This breaking in I think you're referring to is the theory that atomizers and cartos perform better with useage over time then eventually degrade. I think this prime performance after "breaking in" is a myth but that's me. I think once you've got
juice in your kit, the first time you use it it's at its cleanest and purest and should be the best it'll ever be.