Grate the potato.

Optional step (which helps texture) - sprinkle potato with salt, mix well, place in a strainer to drain liquid away. Let drain from 10-30 minutes, as is convenient for you.

Heat a frying pan over low heat until evenly heated. Melt butter in the pan. Add grated potato, and spread out into a patty as thin or thick as you want.

Cook over low heat 5-10 minutes, until potato is mostly cooked through. Then raise the heat to medium, to brown it, about 5 minutes - watching carefully to be sure the butter does not burn.

Flip potatoes in the pan to brown the other side - again, about 5 minutes. If they are not yet fully cooked, now lower the heat and let them cook until done.