We report on continuous high-harmonic generation (HHG) at 1 kHz repetition rate from a liquid-sheet plasma mirror driven by relativistic-intensity near-single-cycle light transients. Through precise control of both the surface plasma density gradient and the driving light waveform, we can produce highly stable and reproducible extreme ultraviolet spectral quasi-continua, expected to correspond to the generation of stable kHz-trains of isolated attosecond pulses in the time domain. This confirms the exciting potential of liquid-sheet targets as one of the building blocks of future high-power attosecond lasers.