We present a new sample of mid L to mid T dwarfs with effective temperatures of 1100 to 1700K selected from the UKIDSS Large Area Survey and confirmed with infrared spectra from X-Shooter/VLT. 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 program, covering 495 sq 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 mass 0.10-0.03Msol and ages 1-10Gyrs. 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, alpha in the range -1