to act the toady, to be a sycophant.
|CUSS 89: Brown (up) [...] Brownie (up) Curry favor with a professor.et al.|
|Dict. of Invective (1991) 61: The sycophant may also be said to brown [someone], to get nose with [someone], to be a brownie (also with allusion to a junior Girl Scout), or to receive brownie points.|
|Alt. Eng. Dict. [Internet] brown (verb) [...] to butter someone up in order to win a favor.|