Having just moved means I don't yet have lots of succulent mature perennial shoots but there are a few divisions that I planted last year who are giving up the goods as well as various wild plants. Also, lots of the seeds I scattered last fall have sprouted.
Lovage makes for a great stock plant. I find it a bit strong to use as a vegetable. Also harvesting dandelions, wild leeks, and parsley for a soup base.
Seeing Results: Fall Sowing
Fall Sowing is a natural way to start germinate seeds that require stratification, such as those that originate from temperate zones, are wild flowers or that self sow. After your garden cleanup, just prepare a seed bed, mark it (this is where I fail, though it does strengthen my identification skills) and sow. If you have mature plants already, cut the ripe seeds heads and either toss them where you want the seedlings to come up or crush and sprinkle them on the ground before composting the rest. It's a fast and easy way to get a headstart on the planting season.
Why do I always overseed? This is called hedging your bets and will require thinning. Turnip Rooted Chevril seedlings. Must be fall sown.
Edible Self Sowers:
The following (not an exhaustive list) have worked for me:
Corn Salad / Mache
Magenta Spreen (aka - giant lamb's quarters)
Dame's Rocket (spring greens are edible)
Chard - Bietina/perpetual spinach is the variety I've let self seed
Bulbing Fennel (regular fennel too I'm sure though I don't grow it)
Sunflowers - especially perennial kinds
Lemon Balm (I like to prevent this as I don't use THAT much)
Violas including sweet violet - a very nice edible this time of year, and Johnny Jump Ups
Garlic and regular chives
Docks and Sorrels
Purslane - including selected varieties like Golden
Turnip rooted chevril
Sweet violets are a great groundcover and have edible leaves in the spring followed by edible flowers. They've grown happily for me in near full shade to near full sun, in various soil conditions.
There are lots more that others have had luck with such as New Zealand Spinach, Celery varieties, more brassicas so have an internet-look at the possibilities for your garden and lots self sow what you like to grow. You'll see that members of the carrot, cabbage, and goosefoot family are very well represented.
I also have various herbs and other vegetables that come back like most of the tomato family relataives - sunberry, tomato, potato, ground cherry - but some people may wish to remove these volunateers if they have disease issues. Vining crops will also often volunteer but as many people grow more than one of the same Species genus (Cucuribit maxima for example), they will result in crosses. Could be fun, and compost squash often grow great.
Many flowers like alyssum and coreopsis also self seed and might be a lovely addition to your sea of self seeded greens as long as you can tell the leaves apart! When managing your self sown border, you can let it have the wild look or selectively thin so that you have patches of the same plant or a few complementary plants. Stripes are nice too. Let the ones at the back develop their seedheads which are often shockingly tall compared to the juvenile leafy rosette. If you include a few taller flowering perennials like echinacea, they'll hide the sometimes unruly looking seed heads of chard and lettuce and prop up the gangly but attractive seedheads of plants like chicory and kale. Some plants, such as mint, you might want to cut back after the bees have their fun but before they set seed. Not only is mint invasive enough as it is but the seedlings are often inferior in taste.
If you have allowed your self sowers to add to the soil seed bank year after year but mulch consistently so that you might see less self sowers, stir up the soil in a patch and see what emerges. Last year, when I removed my vegetable garden and put in a yawn lawn to sell our property, among the grass and clover sprung up a salad bowl of goodness.
Wild Garden Seeds (recently Frank Morton was profiled in Mother Earth Magazine) sells a mix of feral edibles called an insectary mix which would be a great semi-wild border that would produce a nice bounty to boot!