Generating crisp, i.e., one-pixel-wide, edge maps remains one of the fundamental challenges in edge detection, affecting both traditional and learning-based methods. To obtain crisp edges, most existing approaches rely on two hand-crafted post-processing algorithms, Non-Maximum Suppression (NMS) and skeleton-based thinning, which are non-differentiable and hinder end-to-end optimization. Moreover, all existing crisp edge detection methods still depend on such post-processing to achieve satisfactory results. To address this limitation, we propose \MethodLPP, a lightweight, only $\sim$21K additional parameters, and plug-and-play matching-based supervision module that can be appended to any edge detection model for joint end-to-end learning of crisp edges. At each training iteration, \MethodLPP performs one-to-one matching between predicted and ground-truth edges based on spatial distance and confidence, ensuring consistency between training and testing protocols. Extensive experiments on four popular datasets demonstrate that integrating \MethodLPP substantially improves the performance of existing edge detection models. In particular, \MethodLPP increases the Average Crispness (AC) metric by up to 2--4$\times$ compared to baseline models. Under the crispness-emphasized evaluation (CEval), \MethodLPP further boosts baseline performance by up to 20--35\% in ODS and achieves similar gains in OIS and AP, achieving SOTA performance that matches or surpasses standard post-processing for the first time. Code is available at https://cvpr26-matched.github.io.