When using the Air for charging as I am, from what I have seen, when the battery is completely charged the unit stops charging. How does an overcharge work? Do I need an external charger to do that?
Leave it plugged in after the LED stops flashing, at least 4 hours longer, overnight is better.
Chargers not only recharge following the CC/CV routine but have to sense there's a battery to charge. They do this by trying to charge at a very modest level (called 'detection current' in the trade). This same or similar current is also used to 'unlock' protected batteries that have been to discharged and have gone into self protect mode. The unit has to provide a 'trickle charge' to recharge them enough to unlock the protection PCB and allow normal charging after detection, right?
These two currents (often really only one used twice) aren't strong enough to damage the battery (obviously) but that can ensure that 100% of the available chemicals in the cell are available (otherwise they remain unavailable after the first discharge). This is often called a 'forming charge' in the industry.
If you've discharge the new battery (a heavy discharge, like trying to heat, briefly turning it on without heating might not count) even 'just a little to see how it works or smells or how long it takes to heat up' the opportunity for improving it's performance by overcharging is forever lost. Not to worry, just enjoy what you have.
Such a practice won't give many more 'cycles' (charge/discharge) to the battery life (that's mostly set by temperature and charge start and end levels (and how much time it spends in such conditions). Best results thee come from avoiding/limiting 100% charge levels and never discharging deep enough to 'lock out'. It can, however, add capacity to each cycle. And it's easy enough to do one time? Cheap insurance IMO.
OF