Oscillator starts precisely on edge.. ie. synchronous
Any oscillator needs a few swings until it is stable. So your idea isn't directly implementable. What you can do, however, is to synthesize the waveform you need.

Use one of the PWM outputs each Raspi has to generate a high-frequency rectangle wave with variable pulse width and feed that through a low-pass to get the desired waveform as a result. You only need a resistor and a capacitor for that, all else is software.