Burdock (Arctium minus) is a biennial weed, meaning that the plant matures over a two year period. It germinates in the spring, produces mainly vegetative growth the first summer, overwinters as a rosette and the second summer it grows, flowers, sets seed and dies. And it makes a LOT of seeds, which can remain viable in the soil for a long time.
First, remove all second year (tall) burdock stalks before they have had a chance to flower and set seed. Next, any rosettes emerging this spring should be hand-dug while they are still small. There is a long tap root, so you will need to dig along both sides of the root with a spade, deeply enough to get the whole root. It will be difficult to avoid breaking the root.
If the piece of root sends up new leaves, you can try pouring liberal amounts of horticultural vinegar between the leaves so it will run down to the roots. Over the next couple of days, the stems and leaves will die. Pull them out and pour more vinegar directly on the root. Then, if new leaves appear, be diligent about cutting them off the ground. The root will have been weakened and without the leaves to help nourish it, it will eventually die.
If these weeds are covering a large area of your lawn, you may want to try starving them of light. Here’s a link to our previous answer to another inquirer, which describes this method.
Best of luck!