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.