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BRUSSELS — The calls began in December, as the United States prepared to administer its first batches of Covid-19 vaccine. Even then, it was clear that the European Union was a few weeks behind, and its leaders wanted to know what they could learn from their American counterparts.

The questions were the same, from President Emmanuel Macron of France, President Ursula von der Leyen of the European Commission, and Alexander De Croo, the prime minister of Belgium.

“How did you do it?” Dr. Moncef Slaoui, the United States vaccine czar, recalled them asking on the calls. “And what do you think we missed?”