Dadblamed broadleaf weeds

Every day when Cliff and I take our walk, on the way past the field he planted in grass last fall, he'll say something like this:  "I gotta get out here with the hoe and get those dad-gummed broad-leaf weeds."  
Three years ago when a late freeze killed most of the alfalfa in this little plot, weeds took over.  And although Cliff disked the area repeatedly last summer, we would see big plants waving in the breeze, thumbing their noses at us.  

This morning when Cliff said, "I intended to get out here with the hoe and get those dad-blamed broadleaf weeds, but it looks like it ain't gonna happen," I thought, "Maybe he's hinting for me to do it." 
So after planting some trees and tilling the garden this morning, I headed out with my trusty hoe.  From where we took our walk, we'd seen perhaps a half-dozen of the big weeds; I figured it wouldn't take long to get rid of them so Cliff could find something new to talk about on our walks.  

Bonnie and Sir Loin followed me in and started enjoying the new grass; you wouldn't think there were any broadleaf weeds there, would you?  Just lovely, green grass, waving in the breeze. 

Once inside the electric-fenced patch, though, I saw there were dozens... perhaps hundreds... of the dad-blamed weeds  They were just getting established, so they were too small to see from our path.  
The trouble is, when I use the hoe to get rid of them, the root is still there to send up another plant.  We hate to use a broadleaf weed killer, since we inter-seeded clover there last winter.   
It almost makes me want to move to town.