I've been doing some Hochkurz No-Sparge batches lately, and have enjoyed the convenience and ease of experimenting with step mashes when I forgo the sparge. But I am getting about a 15-20% efficiency drop, which leaves me wondering how effective of a partigyle I could try. The idea would be to fill the tun with a second full boil volume of water immediately after draining it of the first batch's sweet wort. Then I would drain the cooler tun about 90 mins later when the first beer has chilled and gone into the fermenter. Is there anything wrong with this plan?
I might be misunderstanding, but what you have decribed sounds to me more like a normal partigyle, not a "no-sparge" partigyle, which sounds to me like an oxymoron. You're essentially using the first runnings for your first beer, and then batch-sparging but using all of the second runnings for the second beer. This is precisely how I have done partigyle in the past, with the exception that after running off the first runnings, I set it aside without mashing out, while I made the second beer to completion. Skipping mashout and continuing right on with the second beer resulted in extended saccharification of the first runnings, allowing this wort which would become a barleywine to be highly fermentable and not poop out when FG hit about 1.030 or whatever. If I'd wanted to have a beer with normal attenuation, I could have included a mashout step by bringing the sweet first runnings (not the mash itself but just the sweet wort) to 170 F to kill the enzymes before continuing on.
There is zero need to extend the mash time for the second beer -- it will turn out thin and watery and kind of lifeless regardless. As long as you did a good mash for the first beer, you can just add more water to make the second beer, stir the mash, wait 20 seconds, then immediately drain it out and continue brewing as normal.
Another nice thing I like to incorporate into my partigyle is to swap a little bit of the two worts, e.g., take a few quarts of the first runnings and blend into the second, then take some of the second to put back into the first, and do this a couple of times, which brings the gravity of the first beer down slightly but brings the gravity of the second beer up quite a bit. Otherwise the second beer inevitably turns out so thin. I theorize that most of the good flavor compounds are retained in the first runnings and don't make it into the second wort, so a little blending can help jack up the second beer without hurting the first one much. Also I like to steep specialty grains in the second wort as well to help jack it up better too. Just a few ounces or up to a pound maybe is all that's really needed to keep the second beer more lively in character.