Modern production systems are increasingly defined by dense networks of multi-tier sourcing dependencies, where localized upstream disruptions can cascade into system-wide collapses. While supply chain resilience has garnered significant managerial attention, we still lack theoretically-grounded, reliable, analytical metrics that can distinguish inherently resilient architectures from fragile ones. This paper addresses this gap by developing a structural resilience framework and a novel metric, defined as the maximum supplier failure rate that a network can sustain while maintaining an aggregate production level. Using node percolation theory and branching processes, we identify four critical structural determinants of resilience: the number of raw materials, the number of finished goods, sourcing requirements, and sourcing influence. Our analysis reveals two distinct regimes: "top hat" architectures, which are characterized by excessive raw materials and high centralization, making them inherently fragile; and "rolling pin" structures, which maintain controlled input/output widths and sparsity, allowing them to absorb non-trivial shocks. To operationalize these insights, we formulate resilience computation as a scalable linear program that approximates cascading failure sizes in large-scale networks with cycles, heterogeneous suppliers, and structural decoupling. Furthermore, we extend our framework to account for exogenous failure correlations, such as those arising from geographic or geopolitical factors that can undermine traditional supplier and input diversification strategies. We validate our theoretical results using multi-echelon supply chain data. These tools can inform network design, supplier diversification, and inventory planning to proactively reduce systemic risk.