The functional interpretation of language-related ERP components has been a central debate in psycholinguistics for decades. We advance an information-theoretic model of human language processing in the brain in which incoming linguistic input is processed at first shallowly and later with more depth, with these two kinds of information processing corresponding to distinct electroencephalographic signatures. Formally, we show that the information content (surprisal) of a word in context can be decomposed into two quantities: (A) shallow surprisal, which signals shallow processing difficulty for a word, and corresponds with the N400 signal; and (B) deep surprisal, which reflects the discrepancy between shallow and deep representations, and corresponds to the P600 signal and other late positivities. Both of these quantities can be estimated straightforwardly using modern NLP models. We validate our theory by successfully simulating ERP patterns elicited by a variety of linguistic manipulations in previously-reported experimental data from six experiments, with successful novel qualitative and quantitative predictions. Our theory is compatible with traditional cognitive theories assuming a `good-enough' shallow representation stage, but with a precise information-theoretic formulation. The model provides an information-theoretic model of ERP components grounded on cognitive processes, and brings us closer to a fully-specified neuro-computational model of language processing.