Apparently a good night's sleep and some strategizing does wonders for a boss attempt. Today I was able to go into Castle Cainhurst and take out Logarius in a single bout.
The key, I think, was being aggressive in phase one. If you can get into melee range and hit him with a couple attacks you'll prevent him from casting some of his spells. It's still not safe - he has some moves that you can't prevent - but it'll push him to the next phase a lot sooner.
The absolute key to making the next phase easier is interrupting him when he starts casting his buff. Luckily he'll be in melee range when he starts, as your damage is what gets him to start casting, and then you just have to hope that he doesn't have his back against a wall. A single charged R2 will break him out of it and you can get off a visceral attack to boot. Then, you can just parry his melee attacks (some are really fast, so you might need to back off and heal occasionally) and dodge when he flies up into the air.
With Logarius down, I finished (maybe I have one more step - I left him in Cainhurst) Alfred's questline and then grinded a bit in Hemwick to get the echoes required for the Gold Areo and the Logarius Wheel (well, I spent my boss echoes on the latter.)
The Wheel takes some getting used to after primarily using the Kirkhammer's rather fast sword. But I'm finding that I can slam the wheel pretty darn hard and even take out multiple enemies.
Next I headed for the Nightmare Frontier. I got thrown off a cliff by Patches (who I seemed to miss when playing Dark Souls) and eventually got through the appropriately nightmarish landscape to face down Amygdala.
And I one-shot him. That's three one-shot bosses on the first playthrough!
Amygdala (pronounced at least by Patches with an emphasis on "Dal" instead of "Myg," as you'd use when talking about the part of the brain) wasn't too hard, mainly because his attacks were relatively easy to predict and dodge. I had maybe one really scary moment when I got stunlocked by his cosmic-fist attack, but basically, testing out my new Logarius Wheel on his arms and head (his legs take like no damage) with a visceral attack in there as well, I was glad to be able to put the Frontier behind me (though there were some treasures I missed.)
So now I think what's left to me is Yahar'gul and the Hunter's Nightmare. I know I can get the key to Upper Cathedral Ward early in the former, and might shoot on over there when I can. I think I'm probably going to save the Hunter's Nightmare for last, just before Gerhman and the Moon Presence (yes, I fully intend to get the "best" ending.)
And I figure I'll get some Chalice Dungeons in there at some point.