egg suck, v. — Green’s Dictionary of Slang

egg suck v.

[suck up v. (1)/obs. SE suck-egg, a foolish person]

(US) to curry favour; as adj., fawning.

1978		S. King Stand (1990) 90: Denninger looked and acted like the kind of man who would [...] bullyrag them around but lick up to his superiors like an egg-suck dog.
1980		Magnum, P.I [CBS-TV] You can kiss those gold bars goodbye that you spent thirty years eggsuckin’ for! [HDAS].