Light carrying orbital angular momentum L⃗, scattered by a rotating object at angular velocity Ω⃗, experiences a rotational Doppler shift Ω⃗·L⃗. We show that this fundamental light-matter interaction can be detected exploiting self-mixing in a vortex laser under Doppler-shifted optical feedback, with quantum noise-limited light detection. We used a low-noise relaxation oscillation-free (class-A) vortex laser, based on III-V semiconductor vertical-external-cavity-surface-emitting laser technology to generate coherent Laguerre-Gauss beams carrying L=ℏl (l=±1,…±4). Linear and rotational Doppler effects were studied experimentally and theoretically. This will allow us to combine a velocity sensor with optical tweezers for micro-manipulation applications, with high performances: compact, powerful ≫10 mW, high-quality beam, auto-aligned, linear response up to >10⁸ rad/s or >300 km/h, low back-scattered light detection limit <10⁻¹⁶/Hz.