An oscillator is just an amplifier with feedback satisfying the Barkhausen criterion, i.e. the losses in the frequency selective feedback path should be less than the amplifier gain (and the phase should be correct).
An oscillator consisting of a frequency selective element and an amplifier will simply start up amplifying the initial thermal noise a few times through the frequency selective feedback network. After a few iterations through the amplifier, the thermal noise at the desired frequency will saturate the amplifier and only a single frequency is produced.
Thus, feeding a single cycle into a blocked amplifier and bringing the amplifier into a linear mode should produce signals with full amplitude and expected phase.
Paul