Building American Energy Security Act of 2023 This bill establishes requirements to accelerate the environmental review of federal agency actions that involve certain energy infrastructure or critical mineral projects, including the Mountain Valley Pipeline, which is a natural gas pipeline located in Virginia and West Virginia. Specifically, the bill modifies requirements under the National Environmental Policy Act of 1969, including by setting maximum timelines for the environmental reviews of federal actions that involve major energy infrastructure projects, allowing project sponsors to sue agencies if the timelines are not met, establishing requirements to accelerate such reviews that involve multiple agencies, and limiting judicial review of agency decisions. Further, the President must designate and periodically update a list of high-priority energy infrastructure projects and prioritize permitting for these projects. The bill also expands covered projects under FAST-41, which qualify for expedited environmental review, to include certain federal agency actions that involve (1) critical minerals, or (2) energy infrastructure projects that require an investment of more than $50 million. In addition, the bill modifies provisions under the Federal Power Act and the Natural Gas Act, including by authorizing the Federal Energy Regulatory Commission to approve permits for certain interstate electric transmission facilities and hydrogen pipelines. Finally, the bill requires federal agencies to take certain actions to complete the Mountain Valley Pipeline.