Fast-track Logistics for Acquiring Supplies in a Hurry Act of 2023 or the FLASH Act of 2023 This bill expands federal contracting and other authorities to support the development and acquisition of products and supplies that are necessary for national public health and health security. Specifically, the Department of Health and Human Services (HHS) may use noncompetitive procedures to procure medical countermeasures (e.g., vaccines), chemical materials and reagents, manufacturing supplies, and similar items that are needed to develop products for national public health and health security. Additionally, HHS may acquire, subject to dollar limits and other conditions, innovative commercial products and services by soliciting proposals and selecting proposals through peer review. Further, the bill expands the activities of the Biomedical Advanced Research and Development Authority (BARDA) to include supporting the design and development of prototypes for products to diagnose, mitigate, prevent, treat, or otherwise respond to biological, chemical, radiological, or nuclear threats, including pandemics or epidemics. If BARDA enters an initial transaction to design or develop such a test or prototype using competitive procedures, BARDA may use noncompetitive procedures for a follow-on transaction to produce the test or prototype.