Atomic Energy Advancement Act This bill establishes various requirements to accelerate the deployment of nuclear energy technologies, such as advanced nuclear reactors. Specifically, the bill sets forth requirements that direct the Nuclear Regulatory Commission (NRC) to efficiently license and regulate nuclear energy activities. The bill also revises the NRC's hiring process to enable the NRC to respond to a severe shortage of candidates or a critical hiring need by using direct hiring for certain positions to carry out NRC's responsibilities and activities in a timely, efficient, and effective manner. It also reduces certain licensing fees charged by the NRC for advanced nuclear reactors. Further, it authorizes the Department of Energy (DOE) to make awards that pay for certain licensing fees. Additionally, the bill directs the NRC to promulgate a final rule that implements a process to facilitate efficient and timely environmental reviews of nuclear reactor applications. The NRC must also develop and implement strategies to enable efficient and timely licensing reviews for, and to support the oversight of, nuclear facilities at brownfield sites, including sites with retired fossil fuel facilities. Next, the bill sets forth requirements for expediting certain licensing decisions for new nuclear reactors that will use a previously licensed design and will be on or adjacent to a site on which a nuclear reactor already operates or previously operated. It also authorizes DOE to establish a pilot program under which DOE may make certain long-term power purchase agreements for power generated by commercial nuclear reactors. Finally, the bill establishes requirements concerning international nuclear energy cooperation and safety.