Can I breed the most productive hen by hatching eggs from the most productive ones?
In theory yes, so long as you maintain genetic variation within your flock of hens (e.g. by bringing in outsider males to prevent inherited diseases spreading), artificial selection will lead to a more productive flock.

However I don't think it would be predictable and not necessarily an improvement every generation, surely productivity would be influenced by lifespan? e.g. is it better to have a hen that lays 100 eggs/month for 24 months, or a hen that lays 120 eggs/month for 12 months? Not necessarily that important, but worth considering.

You would also want to keep more than one hen per generation, maybe 2 chick hens from each of the 3 most productive hens per generation, just in case you selected one that hadn't inherited the productivity traits.

I assume you've read about milk yields. And this looks like it covers the maths of heritability of egg production.