I had tried something similar in my run with mixed results. I made mine out of 2x2's which I think is a big part of my problems. I wanted something that would keep growing up through the mesh as they trimmed the tops off.
First try: built the frames and place them in the run then threw seed in them. The seed got buried with chicken poop/scratched dirt/straw bits. So nothing much grew.
2nd try. Moved the boxes to the garden with a piece of sheet metal under them. Filled with compost and planted the seeds on top, sprouted nicely. Waited till the growth was very strong and should have formed a little mat of roots. Moved the boxes to the run and slid the sheet metal bottom off and just let the open bottom sit on the run dirt. Within an hour they had pulled all the plants up through the mesh and basically killed the growth.
3rd try(this one will have to wait for spring). I will make the boxes out of 2x4's on edge the same as the OP's boxes. I'll start them in the garden with sheet metal under again, but only fill half full of compost so the soil surface is further from the mesh. When they have strong growth I'll move to run and remove the sheet metal. Hopfully the dirt will be far enough down that the plants will only get trimmed and keep growing up through the mesh. When they start to get a bit ragged I'll just lift the box and let them tear up the roots and start a new one.