In this work, we attempt the experimental demonstration of quantum effects in the motion of a macroscopic mechanical resonator with a mass of 33 micrograms, about 3 orders of magnitude above the mass of the heaviest system demonstrated so far in the quantum ground state. We have designed, fabricated, and operated an optomechanical resonator at 3.6 MHz, with an optical finesse of 100,000 and a mechanical quality factor near 100 million, embedded in the 100 mK environment of a dilution refrigerator. We present a fully automatized optical measurement setup, including a filter cavity, a homodyne detector, and various feedback controllers implemented in an FPGA with the custom-developed software PyRPL. We have laser-cooled the compression mode of our mechanical resonator to a mean thermal occupation number of 20 phonons. Cooling is limited by the onset of an optomechanical instability of suspension modes with frequencies below 100 kHz. A custom-tailored digital filter to suppress this instability has enabled us to reach a regime where quantum backaction amounts to about 30 % of the total force noise on the mechanical resonator. For even higher ratios in the future, we present the design of a phononic-crystal input mirror with a reduced Brownian motion displacement noise floor.