There's a pre-planting technique for potatoes which involves giving
them some sunlight before you pop them into the ground. What happens
is the eyes begin to develop; I usually don't wait until they develop
sprouts, but green buds (I've forgotten the name of the technique -
chitting?). I also don't bother cutting them up, since my local
greenhouse has 5lbs bags for less than $5.00 and no S&H charges.

I get a few (2-4) varieties and usually have enough to do a mid-season
sowing once summer gets here.

So what happens for me is that usually within a week I get leaves
poking up. After plants have 8-12 inches showing they get their first
feeding and hilling.