Green’s Dictionary of Slang

(US) police officers engaged in surveillance duties.

1929		Hostetter & Beesley It’s a Racket! 224: dummy—Detective; officer of the law. Other terms—‘Big Eyes,’ ‘Elephant Ears,’ ‘Round Heels.’.
1935		A.J. Pollock Und. Speaks 7/2: Big eyes, a detective.
1949		Monteleone Criminal Sl. (rev. edn) 21: big eyes A detective; a plain cloth officer.
1950		Goldin et al. DAUL 27/1: Big-eyes. (Chiefly rural) A plain-clothesman or detective.