No, you shouldn't dig second year burdock root in fall.
My folks were over for a visit recently, and brought a stack of herb books that they'd bought for me. They do this now and then: some of the books they buy for me are gems, some I already have, some are simply weird, and some are interesting.
One of the books in this lot is very charming in its own way, but rather ignorant in places. I don't know if I'd write a herb book on one year of personal experience -- but at least this book does have that personal experience, even if it's somewhat short.
On to the burdock (Arctium spp.) root. Let me quote:
Harvesting: dig the first- or second-year roots in the late fall before the ground freezes. Check second-year plant roots (those that have produced flowers and seed burrs) thoroughly, since they are susceptible to rot, especially if growing in damp soil.
Burdock is a biennial.
That means that the plant lives for just two years: it dies after it has set flowers and produced seed. That "root rot" you find in seeding burdock plants is not a disease, it's a plant that's given its all and is dying of old age.
It makes as much sense to dig second-year burdock root in fall as it does to dig second-year carrot roots in fall. Neither contain any of the goodness of first-year fall roots, or even second-year spring roots, if dug before the flowerstalk has formed. Both are either rotten through and through in the fall of their second year, or they're very very woody, because they've been holding up a large flowerstalk.
The same goes for all biennial root crops.
Another quote from the same book:
... I taste a piece of this root, and find it isn't bad either. I'm beginning to think there's something wrong with these roots. I thought burdock was supposed to be bitter, but these roots taste fine. I'll have to try gathering some in another place to see if soil makes a difference. ...
There's nothing wrong with your burdock root, and you won't find any difference if you dig burdock root elsewhere.
Burdock root is delicious.
Especially after it's been washed, sliced, and dried. Yum!
Unfortunately digging burdock root is so much work. The plant doesn't like nice earthy places, it likes stones. Lots ov'em. Small ones, big ones, all around, making certain you won't get your shovel all that far down. It's rare to get a whole root when you dig burdock; mostly they just break, about 10-15 cm down.
Planting in nice stone-less soil isn't a solution either, because while garden-grown burdock root is a perfectly fine vegetable, it's not very strong as a herb. Burdock needs the poor dirt, the dry spots, the stony bits, in order to become properly burdocky.
(You can buy burdock seeds to plant as a root crop. Find gobo seeds, that's various Japanese cultivars of great burdock, Arctium lappa. Of course, you can plant your local burdocks, too, if you so like - they all have similar roots.)
Anyway. Digging burdock root is a whole lot of work, and after an afternoon or so of digging, and an evening or three (on the same day they've been dug) of washing, slicing and slapping the root into a dryer, it's all gone in half an hour, once it's properly dry a couple days later.
None of my clients ever get to see my burdock root. No, they'll have to settle for the leaf, which is extremely easy to pick in quantity, is bitter(ish), is very much more diuretic than the root, but otherwise works in exactly the same way as the root does.
If you want to properly dry burdock leaf you'll have to cut the central leaf vein all through the leaf (and possibly large side veins, too), else the leaves will turn black, brown, yellow, light green, and normal green from the center of the leaf outwards. This happens because the leaf is so large. I usually thread them up by one of these cuts, beyond the first pair of larger sideveins, and hang them up to dry. In the dehydrator the leaves form a solid flat mass which makes for slow (and low-quality) drying.
Oh, and the name of this charming book? Corinne Martin, Herbal Remedies from the Wild. (or Earthmagic: Finding and Using Medicinal Herbs). 1991, 2000.