Hoary Cress aka Whitetop

I'm not sure this is in the right area, so please feel free to move it.

I've spent hours and hours trying to figure out what to do about whitetop in my yard.  The weed department is willing to spray, but this yard hasn't seen chemicals since 2004.

Cutting each plant and spraying the cut end with vinegar seems to work, but I'm looking at aproximately 300 square feet of solid growth and it spreads faster than I can cut and spray.

I've searched (without luck - or maybe just the wrong words) trying to figure out what this whitetop is trying to tell me about the soil.  Does anyone here know?  I've never had any soil testing done.

This 'lawn' does now have lots of dandelions, yarrow, canada thistle and a decreasing stand of quack grass. And lots of other 'weeds' I haven't figured out yet.  I cut it with the highest setting of my mower and (according to the conventional neighbors) not nearly often enough.  And I love it.

Can anyone help me figure out what to do with the whitetop?

Thank you in advance!

Lots of info in this article.
http://www.wildflowers-and-weeds.com/weedsinfo/Cardaria_spp.htm
Uh oh. It spreads by seed AND RHIZOME!
But you can eat it.
Hand pull it. Ha. It's got rhizomes!
Graze cattle or horses on it. How big is your yard?

OMG!!  How did I miss that?

THANK YOU!  THANK YOU!  THANK YOU!!

Looks like I'll be trying whitetop for dinner tonight!

The affected area is about 50' x 50'.

Thank you so much!