Equal Access to Green cards for Legal Employment Act of 2023 or the EAGLE Act of 2023 This bill modifies requirements related to employment-based visas and addresses related issues. The bill increases the per-country cap on family-based immigrant visas from 7% of the total number of such visas available that year to 15% and eliminates the per-country cap for employment-based immigrant visas. The bill establishes transition rules for employment-based visas such as (1) reserving a percentage of EB-2 (workers with advanced degrees or exceptional ability) and EB-3 (skilled and other workers) visas for individuals not from the two countries with the largest number of recipients of such visas, and (2) allotting a number of visas for professional nurses and physical therapists. The bill imposes additional requirements on an employer seeking an H-1B visa, such as prohibiting (1) an employer from advertising that a position is only open to H-1B applicants or that H-1B applicants are preferred, and (2) certain employers from having more than half of their employees as nonimmigrant visa workers. The Department of Labor shall create a publicly available website where an employer seeking an H-1B visa must post certain information about the open position. The bill also expands Labor's authority to review and investigate H-1B applications for fraud or misrepresentations. The bill also allows certain non-U.S. nationals ( aliens under federal law) to obtain lawful permanent resident status if the individual (1) is in the United States as a nonimmigrant, (2) has an approved immigrant visa petition, and (3) has waited at least two years for a visa.