Electrical – MOSFET to connect two batteries in series

I think what they're trying to do here is 'trick' the phone's battery charging intelligences. Li-On batteries are very touchy and have somewhat complex charging strategies. It all boils down to determining something called State-Of-Charge (SOC). SOC is just a percentage in the end, but arriving at the SOC number relies on a large number of factors that are not always easy to read and sometimes must be indirectly inferred. For instance, let's assume that you have a cell phone with a Li-On battery that is 3.7V and 1000mAH. We'll start with it being fully charged, so we know SOC is 100%. As you use your device you're drawing current out of the battery and the battery's voltage will drop - eventually. By measuring the current and monitoring the voltage you can guess what the SOC is. One problem is that the voltage isn't terribly useful in determining SOC because it doesn't change very much until the battery is nearly empty - that is NOT something you want to do to a Li-On battery. So you're mainly relying on the current.

So your SOC is being estimated throughout the usage. It gets low - 50% maybe - so you plug it in to charge. While it's charging, it monitors the charge current and battery voltage to determine when SOC is 100% again. Only, due to errors in measurement it says that the charge is complete when SOC is actually only 95%. Now your phone thinks 95% is fully charged - and it remembers this for future reference because it doesn't want to over-charge the batteries (this is also very bad). So essentially it's trying to read when the battery is full by measuring what goes in and guessing where that puts the SOC based on past results.

The errors aren't large so during normal charge/discharge usage you won't notice a problem. But sometimes the errors can stack up and your phone thinks its fully charged when it has little or no charge - it goes straight from full to empty and due to the incorrect SOC calculation the phone won't try to charge the battery more because it doesn't want to damage it.

In these cases you have to reset the SOC state. I have a Droid Incredible 2 and I've done it by removing the battery and holding the power button for 30 seconds, then putting the battery in and charging the phone while its off. This always fixes the issue where the battery thinks its full but drops down to something like 10% very quickly and the issue where it thinks its at 10% but has much more charge left.

The strategy outlined in your post is obviously trying to recalibrate the SOC or trick the algorithm somehow. Having never developed a charger that relied on SOC I can't say whether it will work but it seems like a lot of effort for a questionable amount of benefit. If your battery is acting really funny try what I suggested first.

If this is for mass production, then no, you cannot do this. It is never recommended to parallel primary (non-rechargeable) batteries. The reason is that if a charged one is connected to a discharged one, the current will flow into the discharged one, and charging primary cells is a big safety no-no.

If you can devise a circuit method to prevent charging of the cells under all circumstances, then you can do it. (For example, maybe you can accept the voltage drop of a Schottky diode). But otherwise, use a higher capacity button cell. There are many varieties out there.

If you are just goofing around at home, then by all means go ahead. But you should make a point of removing both batteries before you replace them. You might consider adding a resistor in series with each battery to limit the equalization current, just in case you accidentally put a fresh one in parallel with an old one.