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Real-world applications frequently seek to solve a general form of the Entity Matching (EM) problem to find associated entities. Such scenarios include matching jobs to candidates in job targeting, matching students with courses in online education, matching products with user reviews on e-commercial websites, and beyond. These tasks impose new requirements such as matching data entries with diverse formats or having a flexible and semantics-rich matching definition, which are beyond the current EM task formulation or approaches. In this paper, we introduce the problem of Generalized Entity Matching (GEM) that satisfies these practical requirements and presents an end-to-end pipeline Machop as the solution. Machop allows end-users to define new matching tasks from scratch and apply them to new domains in a step-by-step manner. Machop casts the GEM problem as sequence pair classification so as to utilize the language understanding capability of Transformers-based language models (LMs) such as BERT. Moreover, it features a novel external knowledge injection approach with structure-aware pooling methods that allow domain experts to guide the LM to focus on the key matching information thus further contributing to the overall performance. Our experiments and case studies on real-world datasets from a popular recruiting platform show a significant 17.1% gain in F1 score against state-of-the-art methods along with meaningful matching results that are human-understandable.