Prioritizing Veterans Access to Mental Health Care Act of 2017 This bill amends the Veterans Access, Choice, and Accountability Act of 2014 to make a veteran seeking mental health care in a non-Department of Veterans Affairs (VA) entity eligible for such care if the veteran provides a written statement (which may be sent electronically or made in person or by fax) that the veteran seeks mental health care from the VA and has not received timely and adequate mental health care through a VA facility or health care provider. The VA shall give health professional hiring priority to those health professionals for which there is the greatest need, such as psychiatrists, psychologists, and other mental health care providers.