Discharging one side of a capacitor?

It is physically possible for there to be more electrons on one side of a capacitor without there being a corresponding number of holes (absences of electrons) on the other side. In fact, your proposed configuration of two capacitors and a battery would do that — but by a very, very small amount — about the same amount as if you cut a single capacitor in half and spread the plates apart to the same locations, then connected the battery.

This effect, which applies to any conductor, not just capacitor plates, is called self-capacitance, as opposed to mutual capacitance. It is defined in the same way as capacitance,

$$C = \frac{q}{V}$$

— but it is immensely smaller for a given physical size. The amount of charge 1.5 volts — or 9 volts or 240 volts — can push into such a capacitor is so small that it has a negligible effect in typical circuits — we do not bother to think about it.

(It is also true that there is some amount of (mutual) capacitance between the unconnected ends of the two capacitors. Every pair of conductors is a capacitor, but they're usually bad ones with small area and large plate separation! Both self-capacitance and mutual capacitance contribute to how much charge you can stuff into a conductor for a given voltage.)

In electrostatic systems, working with kilovolts and up, the effects of self-capacitance can become significant. If you walk across a carpet and touch a CMOS IC, destroying it, what was the immediate source of the energy at the discharge? It was your body having a net positive or negative charge. The opposing charge was left behind on the carpet. The self-capacitance is the ratio between that amount of carried charge and the voltage between you and the carpet. (Where did the large voltage come from? Separating the “plates”. Where did the initial charge transfer come from? The triboelectric effect.)

A physical example of essentially an “only one side charged” capacitor is a Van de Graaff electrostatic generator. The sphere on top is one plate; the entire surroundings including the Earth (assuming the generator is grounded, as it usually would be) is the other, but the Earth is so much bigger that the charge imbalance is insignificant for it but very significant for the sphere.

No.

The charge on a capacitor is defined by the voltage difference between the two plates, the geometry of the plates, and the chemical properties of the dielectric.

That is.. the charge is between the plates, across the dielectric, not on the plates.

You need to understand it is the presence, or absence of electrons on one plate that drive away or attract electrons on the other plate. You can't change one without changing the other.

As such, the concept of removing charge from one plate is incorrect.

If you remove electrons from the negatively side of the capacitor, the voltage across the plates would drop, as would the charge in the entire capacitor, not just that side of the capacitor.

In fact, the only way to remove the electrons is to change the applied voltage across the capacitor. So we just went round in a nice circle. This is of course what we do all the time when we discharge a capacitor, we apply zero volts across it.

EDIT

There is one way you could achieve what you suggest and that is to use actual plates in a capacitor configuration. Charge them up then disconnect them from the source and then separate the plates. Both plates would still be "charged". You could then discharge one of them to ground and then put them back together. You would then have an unbalanced capacitor. Of course, as soon as you hooked it up to anything, it would immediately try to rebalance itself.