I have been digging them out and have dug down about 1 - 2 feet and gotten some weeds out, but some have broken off with some of the root still in the ground. Is there something I can put on the remaining root to kill it without killing the lawn?
There are selective postemergence herbicides such as 2,4-D that can kill burdock; check the label to make sure it's listed. Read and follow all label directions.
https://www.gardeningknowhow.com/edible/herbs/burdock/controlling-burdock-weeds.htm