@ben-jammin: I'm not still completely sure if I understand what exactly you mean. 10 HA's, ok, how many of them would come from capital? Around 5 I suppose, so maybe 3 whips? Do you build a granary? Could you play a game and post a save? I'll try with minimal whipping.
Can't playright now. There'd certainly be a granary in the capital, built well before HBR. Ideally there'd be a granary in all three cities but depending on how late you get your third city up and running it can do without. Capital for sure gets a barracks, again whether you build one in the other cities depends on how long they take to get founded and grow vs. how fast you can tech HBR and archery.
You want at least 3 workers dedicated to chopping during this phase, which means you ideally want 4+ workers total so you can have at least one roading towards the target. Ideally you start with something like this (in all cities, but capital will of course be 1 siwe bigger than the others):
2 turns before growth into unhapiness start producing an HA, and worker moves onto a forest. 1 turn of production later you're 1 turn to growth and able to whip with ~25 hammer overflow. Whip, the next turn you regrow, the turn after your chop completes the HA. The turn after you should have enough hammers in the next HA for another 2-pop whip and be close to growth, worker has moved to the next chop, and you've produced 4 HAs in 6-7 turns (a bit slower outside the capital where you have less tile production between whips to finish the chopped units and get enough hammers into the next one for a whip).
You obviously don't stop at those 10 HAs though. You'd need at least twice that if the AI has started to produce metal units. But once your forests are chopped it obviously starts taking longer between whips. With the overflow from the last whip usually being too great to 2-pop whip again, and no forest chop to be timed to finish the in-between HA means you slow build it. And it's at this point that having lots of mines available and micromanaging your tiles to avoid growth until the right turn becomes extra important.
But really on Immortal you don't have to worry much about micromanagement and growing into unhappiness for a few turns here and there isn't going to make your rush anyless effective. If you've beelined HBR after worker techs, have some commerce (in this case, all from cottages), and are chopping+whipping aggressively you should have no problem crushing your target with numbers even if they hooked up metal that you can't immediately pillage. Sound tactics are of course also important, luring AI units, pillaging roads and/or metal, skipping well defended cities initially (for every city with 4+ defenders there should be 1 or 2 with no more than two defenders). Also a good road network for your reinforcements and efficient use of promo-healing.