Characterizing the differential excision of mRNA is critical for understanding the functional complexity of a cell or tissue, from normal developmental processes to disease pathogenesis. Most transcript reconstruction methods infer full-length transcripts from high-throughput sequencing data. However, this is a challenging task due to incomplete annotations and the differential expression of transcripts across cell-types, tissues, and experimental conditions. Several recent methods circumvent these difficulties by considering local splicing events, but these methods lose transcript-level splicing information and may conflate transcripts. We develop the first probabilistic model that reconciles the transcript and local splicing perspectives. First, we formalize the sequence of mRNA excisions (SME) reconstruction problem, which aims to assemble variable-length sequences of mRNA excisions from RNA-sequencing data. We then present a novel hierarchical Bayesian admixture model for the Reconstruction of Excised mRNA (BREM). BREM interpolates between local splicing events and full-length transcripts and thus focuses only on SMEs that have high posterior probability. We develop posterior inference algorithms based on Gibbs sampling and local search of independent sets and characterize differential SME usage using generalized linear models based on converged BREM model parameters. We show that BREM achieves higher F1 score for reconstruction tasks and improved accuracy and sensitivity in differential splicing when compared with four state-of-the-art transcript and local splicing methods on simulated data. Lastly, we evaluate BREM on both bulk and scRNA sequencing data based on transcript reconstruction, novelty of transcripts produced, model sensitivity to hyperparameters, and a functional analysis of differentially expressed SMEs, demonstrating that BREM captures relevant biological signal.