Gaseous Bose-Einstein condensates (BECs) have become an important test bed for studying the dynamics of quantized vortices. In this work we use two-photon Doppler sensitive Bragg scattering to study the rotation of sodium BECs. We analyze the microscopic flow field and present laboratory measurements of the coarse-grained velocity profile. Unlike time-of-flight imaging, Bragg scattering is sensitive to the direction of rotation and therefore to the phase of the condensate. In addition, we have non-destructively probed the vortex flow field using a sequence of two Bragg pulses.