I would not let Burdock go to seed - it makes burrs that are very prickly (the inspiration for Velcro, I've been told) and stick to dogs and clothes and many things. The first part of the plant is pretty, and the
root is edible, so if you don't have too many, just be sure to cut the stalks when they pop up.
You can toss them in the
chicken pen. You'll have to do this repeatedly because the plant really, really wants to make seeds. Still, you can think of it as a nutrient accumulator, using that tap root to bring minerals up to the surface. You need to watch it more closely than comfrey (famous
permaculture nutrient accumulator, which is usually sterile and doesn't spread by seed). Just keep chopping and dropping!
When I moved into my place in Wisconsin, back in 2000, I had a few of these that I saw in the spring and didn't identify. They became a huge pain for my long haired German Shepherd dog. I didn't know about the chop-n-drop strategy then, so I would try to dig them up (a once and done strategy). It was exhausting and I was not very successful. Much easier to carry a hori knife with you and just chop them when you see them.