Accelerating Deployment of Versatile, Advanced Nuclear for Clean Energy Act of 2023 or the ADVANCE Act of 2023 This bill sets forth provisions to develop and deploy advanced nuclear fuel for the United States and certain allied countries, restrict the possession or ownership of enriched uranium from Russia or China, clean up hazardous land, and establish related requirements. Specifically, the bill provides incentives for developing and deploying new nuclear technologies, such as reduced licensing fees and prize awards for deploying such technologies. It also extends through 2045 the indemnification policy under the Price-Anderson Act that limits liability related to the nuclear industry. It also requires the Nuclear Regulatory Commission (NRC) to (1) develop a process that enables timely licensing of nuclear production facilities or utilization facilities at brownfield sites, and (2) establish an initiative to enhance preparedness and coordination with respect to the qualification and licensing of advanced nuclear fuel. NRC may hire specialized staff without regard to civil service laws to address its critical licensing or regulatory oversight needs. The NRC must also coordinate certain international nuclear activities and may establish an International Nuclear Reactor Export and Innovation Branch within the Office of International Programs. The bill allows certain foreign entities to receive licenses under the Atomic Energy Act of 1954 for nuclear utilization facilities as specified by the bill. The bill also allows the Environmental Protection Agency to conduct activities to assist in cleaning up certain abandoned mining sites on tribal lands that are hazardous.