I kept seeing complicated recipes that would be more trouble than they were worth. I eventually worked out a very simple one that I find satisfactory.
Makes one large serving.
To make multiple servings, I’d use a larger pan, and I’d try making the patty thicker (cooking covered should cause the part not in contact with the hot pan to still cook by steaming).
1 medium russet potato
1 pat butter
1T olive oil
salt and pepper
Scrub, remove eyes and anything else you don’t want to eat, and coarsely grate the potato.
Pre-heat a cast-iron skillet of suitable size (I find 8″ works well for one serving). Must have cover. On my particular stove the cooking setting is #4 on the left rear burner, and I pre-heat at 6 or 8 because I’m impatient. I might call that “medium” heat?
Add butter to pan. It should sizzle a bit, right away, but not burn. Also add olive oil.
Dump the potato shreds into the pan, and arrange them flat over the entire bottom of the pan. Pressing down a little doesn’t hurt.
Cover the pan and cook for 5-7 minutes (depends on heat, size of potato, etc.).
Flip the patty of potato. It should stick together well, not stick to the pan, and the bottom should be medium brown to golden. Push down all over.
Grind some pepper and shake some salt over the cooked side.
Cover the pan again, and cook this side for 5-7 minutes. If you think it ends up too moist, omit the cover for some or all of the second cooking period.