Potato tubers are not actually roots. They are modified stems. So the surest way to force more potatoes is to “hill” them. In the commercial fields this is done with a huge tractor raking soil from in between planting rows and piling it up on the plants. You essentially bury the plants stem as it grows taller. Then the buds on the stem will push out stolons (horizontal underground stems.) these will terminate in tubers, aka: potatoes!
Source: did potato disease research for my PhD.
Additional edit: loose/sandy soil is critical. Too dense of soil and your tubers can’t expand well.
Warning: I am not a beet expert. But I believe beets are actual roots. Just like carrots. And I think you only get one beer per plant? Burying the stem would just make it harder for new leaves to come up.
Potatoes are pretty unique in this sense. Even sweet potatoes are not the same.
The leafy top is called a haulm and on commercial farms the harvester has a header that removes the haulm before the main part of the harvester scoops up the potatoes. Anyone who's played Farming Simulator is familiar with these machines, such as the Ropa Panther 2.
Nitrogen grows green leaves, Phosphorus and Potassium help with root growth. Those are the 3 elements are the ones you are trying to keep in balance. Or more in one direction depending on the plant.
Obviously there is a bit more to it then that, but that's the general idea.