Recently, full duplex (FD) has been studied in 5G LTE millimeter wave (mmWave) cellular communications for New Radio in 3GPP releases 15-17. FD allows bidirectional transmission over the same resources and has the potential to reduce latency and double spectral efficiency. Self-interference (SI) is the primary drawback. SI can be several orders of magnitude greater than the received signal power, saturate the analog-to-digital converters (ADCs) and degrade communication performance severely. Massive mmWave antenna arrays may provide enough degrees of freedom for spatial multiplexing and SI suppression. In this paper, we design spatial beamformers for the phased arrays already built into the FD basestation/relay to extend mmWave coverage to a single user. We propose alternating projections to design the precoder and combiner to maximize the sum of the uplink and downlink spectral efficiencies while bringing SI below the noise floor. Our contributions include (1) hybrid analog/digital beamformer design algorithm to cancel SI in the analog domain to avoid ADC saturation and in the digital domain on each subcarrier; (2) full-digital beamformer design algorithm; and (3) analysis of spectral efficiency, energy efficiency and outage probability. In simulation, the proposed algorithms outperform beamsteering, singular value decomposition, angle search, and half-duplex techniques.