We laser cool a centimeter-scale torsion pendulum to a temperature of 10 mK (average occupancy of 6000 phonons) starting from room temperature (equivalent to ~10^8 phonons). This is achieved by optical radiation pressure forces conditioned on a quantum-noise-limited optical measurement of the pendulum’s angular displacement with an imprecision 13 dB below that at the standard quantum limit (SQL). The measurement sensitivity is the result of a novel ‘mirrored’ optical lever that passively rejects extraneous spatial-mode noise by 60 dB. The high mechanical quality (10^7) and quantum-noise-limited sub-SQL measurement imprecision demonstrate the necessary ingredients for realizing the quantum ground state of torsional motion — a pre-requisite for mechanical tests of gravity’s alleged quantum nature.