Mr. Carvalho, 53, has led Miami-Dade, the nation’s fourth-largest district, for nearly a decade and has a remarkably successful record. The high school graduation rate has increased to 80.7 percent in 2016-17 from 60.5 percent in 2007-8. Mr. Carvalho grew up poor in Lisbon, Portugal, and was the first in his family to graduate from high school. At the age of 17, he flew to New York on a visa that he then overstayed. In New York, he washed dishes and worked as a busboy in restaurants. He made his way to Florida, where he worked in construction. He has said that at one point he spent a month being homeless, sleeping in a friend’s U-Haul truck. In a restaurant where he worked as a waiter, he met Representative E. Clay Shaw, a Republican congressman from the area, who helped him gain a student visa. Mr. Carvalho has spent his entire career in the Miami-Dade school system, starting as a teacher at Miami Jackson Senior High, where he taught physics, chemistry and calculus. He went on to become an assistant principal, a lobbyist for the district, and an associate superintendent. He took over the district from Rudy Crew, himself a former New York City schools chancellor. Mr. de Blasio’s wife, Chirlane McCray, was also involved in the decision, Mr. Phillips said. Ms. McCray has played a role in many of Mr. de Blasio’s important hiring choices and recently has spoken along with the mayor at the news conferences to announce them.