We present a new sample of mid-L to mid-T dwarfs with effective temperatures of 1100-1700 K selected from the UKIDSS Large Area Survey (LAS) and confirmed with infrared spectra from X-shooter/Very Large Telescope. This effective temperature range is especially sensitive to the formation history of Galactic brown dwarfs and allows us to constrain the form of the sub-stellar birth rate, with sensitivity to differentiate between a flat (stellar like) birth rate and an exponentially declining form. We present the discovery of 63 new L and T dwarfs from the UKIDSS LAS DR7, including the identification of 12 likely unresolved binaries, which form the first complete sub-set from our programme, covering 495 square degrees of sky, complete to J = 18.1. We compare our results for this sub-sample with simulations of differing birth rates for objects of masses 0.10-0.03 M⊙ and ages 1-10 Gyr. We find that the more extreme birth rates (e.g. a halo type form) can likely be excluded as the true form of the birth rate. In addition, we find that although there is substantial scatter we find a preference for a mass function, with a power-law index α in the range -1 <α < 0 that is consistent (within the errors) with the studies of late T dwarfs.