In 1929 in Savannah, Ga., Edna Sutton dreamt of being a nurse. But, as a young black woman, she had few opportunities in her hometown, which was still ruled by Jim Crow laws. The only nursing she could do was visiting the Negro settlements on the outskirts of the city, administering whatever homespun help she could to ailing patients in clapboard shacks. So, she headed to New York, where there was a widespread nurse shortage. Amidst the tuberculosis epidemic, young white women were leaving the field in droves for less dangerous opportunities. The city recruited young black women from the...