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Zinsser Cover Stain is a solid option. It's oil based and not moisture vapour permeable but resists the wood staining the primer, I use it on any wood that has been stripped and most new wood. Zinsser 123+ often works well enough on new hardwood as long as there are no problems with staining which varies from one piece to another in my experience.
I would normally add an undercoat matching the manufacturer of the eggshell on top of the primers I've mentioned above.
Can't find it in here but what's the goto for priming bare wood windowsills (hardwood) and architrave (pine) before eggshell?