Sow Thistle - how do I get rid of it?!?

I have sow thistle growing like crazy through one of my fruit tree guilds that I was hoping to revamp this year. I started to try to dig them up this morning and spent about an hour on one little patch, attempting to follow the roots FEET in each direction. My soil is really bad clay so digging the stuff up is nearly impossible, and the roots just keep breaking (from what I understand this just creates more plants and a bigger problem).
My question is what is my best method of attack? Keep trying to dig them out? Cardboard with lots of mulch? Pick the leaves as often as possible to try to exhaust the plant? Plant something else to try and out compete it?
I've included pictures for reference and in case I have not identified it correctly as sow thistle.
Thanks!

Eat them? http://www.eattheweeds.com/sonchus-sow-thistle-in-a-pigs-eye-2/

Yes, I was also going to say, eat them!

If you've got "really bad clay soil" and these sow-thistle roots are going all over the place down inside of it, that sounds like it might be a good thing... So pulling the above-ground off on a regular basis along with a bit of root might reduce size and spread of the sow-thistle and allow you to get other things established. I'm not sure, but sowthistle might be one of those weeds that loves to occupy "disturbed soil", and might reduce in vigor as you get other perennials covering your soil, and no further digging over time. On our canal sides, the sowthistle is one among many things growing there, and doesn't try to take over.