Yes! You can plant a sprouted potato and grow more potatoes. You will actually get several potato plants and ultimately a bunch of new potatoes from just one sprouted potato if you do it right.
First, prep your garden bed by mounding soil into rows. Potato plants grow best when you mound the dirt. Mound the soil in your garden by scooping the dirt on either side of the row you create and piling the dirt up in the middle. Mound the soil up to between 8" and 12" tall.
A potato sprouts from the eyes on the potato. Each sprouted eye can be cut and grown into a plant. Separate the sprouts by cutting away each group of sprouts, leaving a small amount of potato attached to the sprout.
You'll want to plant your potato sprouts within 2-3 days of cutting. Potato sprouts should be planted cut-side down, sprout-side facing up. You'll want to plant each sprout 3-4" below the surface of the soil. Plants should be spaced out at least 12" apart.
Planted potato sprouts will take about a week to push through the soil and open up their leaves. Keep the plants watered and make sure they get plenty of sun. Continue to water and weed around your potato plants. Potatoes need about 3 months to grow and produce a harvest.
You'll know that your potato plants are ready to be harvested when the visible plant dies off. Harvesting potatoes is like digging for treasure. Sift through the mounded dirt until you find your potatoes. You'll likely get a variety of sizes.
Do not wash your harvested potatoes. You want to keep potatoes dry to avoid rot. To clean off the dirt, use a dry, soft vegetable brush. Store them in a cool dry place.