Green’s Dictionary of Slang

1. (US) a piercing stare.

1953		E. Reid Shame of N.Y. 89: Examiners are told to cast their hawk eyes about for signatures without dates, without E.D. and/or A.D. indicated, signatures of persons unenrolled.
1960		I. Freeman Out of the Burning (1961) 204: He had the hawkeye on me so fast.

2. (Aus. prison) a prison officer.

1990

Tupper & Wortley Aus. Prison Sl. Gloss. 🌐 Hawk eye. An officious prison officer.