You're wondering, "Can I plant these sprouted onions and regrow a new, fresh onion?" In short the answer is, YES! You can plant a sprouted onion and grow a new one.
Peel the onion layer by layer until you reveal separate plants. Separate each onion plant by carefully cutting it apart at the root. Now it's time to plant your onion sprouts. You will need to plant them in full sun.
Onion plants don't need to be buried very deep. You should plant them at least 4" apart so the bulbs have room to grow. Planted onion sprouts will take about 3 weeks to look perky. Keep them watered and plant them in a place that gets plenty of sun.
Onion plants prefer to be watered in cycles rather than every day. So you can water them about once a week unless the soil and plants start to visibly look too dry. Onions grow best in fertilized, well-drained, somewhat acidic soil and they like full sun.
Your onions should grow for about 3 months to reach optimum size which is when the bulb reaches the size of a baseball. You do not want your onion plant to flower. If it does, then you'll want to harvest that plant. If your onion does not flower then you can let it grow up until the first frost.
Harvest by pulling the onion out of the ground and then brushing them to remove dirt. Do not wash onions unless you are planning to eat it right away. You want to keep the onions dry to avoid rot. You can "cure" your onions for long-term storage by letting them sit in a warm place to dry up.
Once cured store your onions in a cool, dry place. Cured onions, depending on the type and size, can be stored for as long as 10 months!