The electric vehicle (EV) battery supply chain will face challenges in sourcing scarce and expensive minerals required for manufacturing and in disposing of hazardous retired batteries. Integrating recycling technology into the supply chain has the potential to alleviate these issues; however, players in the battery market must design investment plans for recycling facilities. In this paper, we propose a multistage stochastic optimization model for computing minimum cost recycling capacity decisions, in which retired batteries are recycled and recovered materials are used to manufacture new batteries. To construct a realistic and high-fidelity model, we transform EverBatt, a leading evaluation framework for battery recycling cost and environmental impact, into a prescriptive decision-making tool for determining optimal investment strategies. Our model is a separable concave minimization subject to linear constraints, a class for which we describe a finitely convergent global optimization algorithm based on piecewise linear approximation that solves up to 14x faster than comparable algorithms. We propose an equivalent reformulation of the model that reduces the total number of variables by introducing integrality constraints. The reformulation can also be solved by our global algorithm with drastically reduced solve times. To produce a set of operational scenarios, we generate time-series projections for new battery demand, retired battery supply, and material costs, leveraging state-of-the-art econometric models for critical metal prices and EV demand. We apply our model to compute optimal investment plans and quantify the impact of optimal decision-making on the supply chain. The optimal solutions show that effective investment in recycling can reduce battery manufacturing costs by 22% and environmental impacts by up to 26%. We measure the operational benefits of different recycling technologies, including material recovery efficiency and remanufacturing flexibility, and demonstrate how high uncertainty in material costs and EV adoption motivates greater investment in recycling capacity. Finally, we analyze the impact of policy instruments to reveal key insights for policy makers: while recycling capacity grants and production credits both effectively encourage domestic investment in recycling, production credits are more cost effective under smaller policy budgets, whereas grants become the more cost-effective instrument as total policy expenditure increases.