Hello Joyce, and thank you for contacting us at the UVM Extension Master Gardener Program with your question.

Burdock is a persistent and frustrating plant to have to deal with!  You are correct in that it is a biennial, so we need to consider the two years of its lifecycle.  It grows from seed in the first year, then produces flowers and seed in its second year.  While you can make some progress in defeating it by mowing over it, as you have seen, it will take several years of persistence because the seeds already in the ground can survive 2-10 years.

It can be tamed, and the key is to remove the plant - or cut it down - before the seeds become viable.   Hand pulling the plant in its first year is ideal, and most practical for smaller areas.   Hand cutting it or mowing it before seeds are produced in its second year will prevent a new crop of seeds from being spread. It does not usually survive tilling, though that might only take care of only the current year's plants. 

Here are some additional resources on the topic from other university extensions that might provide additional insight for you:

https://extension.psu.edu/weed-management-in-pasture-systems

https://www.ndsu.edu/agriculture/ag-hub/publications/identification-and-control-invasive-and-troublesome-weeds-north-dakota

We've also consulted with Ann Hazelrigg, Plant Pathologist at UVM, and she acknowledged that burdock can also be controlled by applying Roundup or 2,4-D just to the plant.   Even then, it can take several years to get the weed under control, as you still might have seeds coming up from previous years.

I wish you luck with eliminating the burdock!