Large artificial intelligence (AI) models have garnered significant attention for their remarkable, often "superhuman", performance on standardized benchmarks. However, when these models are deployed in high-stakes verticals such as healthcare, education, and law, they often reveal notable limitations. For instance, they exhibit brittleness to minor variations in input data, present contextually uninformed decisions in critical settings, and undermine user trust by confidently producing or reproducing inaccuracies. These challenges in applying large models necessitate cross-disciplinary innovations to align the models' capabilities with the needs of real-world applications. We introduce a framework that addresses this gap through a layer-wise abstraction of innovations aimed at meeting users' requirements with large models. Through multiple case studies, we illustrate how researchers and practitioners across various fields can operationalize this framework. Beyond modularizing the pipeline of transforming large models into useful "vertical systems", we also highlight the dynamism that exists within different layers of the framework. Finally, we discuss how our framework can guide researchers and practitioners to (i) optimally situate their innovations (e.g., when vertical-specific insights can empower broadly impactful vertical-agnostic innovations), (ii) uncover overlooked opportunities (e.g., spotting recurring problems across verticals to develop practically useful foundation models instead of chasing benchmarks), and (iii) facilitate cross-disciplinary communication of critical challenges (e.g., enabling a shared vocabulary for AI developers, domain experts, and human-computer interaction scholars). Project webpage: https://gaurav22verma.github.io/vertical-systems-with-large-ai-models/