This paper considers the ergodic weighted sum rate (WSR) maximization problem for an underlay cognitive radio MISO broadcast channel, where a secondary network, consisting of a base-station with M transmit antennas and K single-antenna secondary users (SUs), is allowed to share the same spectrum with a primary user (PU), under an average transmit sum power (ATTP) constraint P av and an average interference power (AIP) constraint on the PU. We show that the ATTP constraint is always active, and as P av → ∞, the ergodic WSR approaches infinity similar to the conventional non-CR network case. A low-complexity suboptimal beamforming scheme (called partially-projected regularized zero-forcing beamforming `PP-RZFBF') with a closed-form beamformer is proposed. Due to the non-convexity of PP-RZFBF scheme, a large system analysis is conducted in the limit as M and K approach infinity with a fixed finite ratio r = K/M. We derive deterministic limiting approximations for the PP-RZFBF problem which enables us to determine asymptotically optimal beamformers for PP-RZFBF. Numerical simulations illustrate that the asymptotically optimal beamformers turn out to be quite effective even for small M, K.