We propose a conditional random fieldbased method for supertagging, and apply it to the task of learning new lexical items for HPSG-based precision grammars of English and Japanese. Using a pseudo-likelihood approximation we are able to scale our model to hundreds of supertags and tens-of-thousands of training sentences. We show that it is possible to achieve start-of-the-art results for both languages using maximally language-independent lexical features. Further, we explore the performance of the models at the type- and token-level, demonstrating their superior performance when compared to a unigram-based baseline and a transformation-based learning approach.