This scrutiny of technical minutiae has turned the European Union into a regulatory superpower, allowing it to help set norms and standards used around the world. But that tight focus has crippled its ability to grapple with big issues or to engage with many ordinary people. As the British vote showed, many people feel no connection with what began as an idealistic peace project after World War II, but is now widely viewed as a meddling and undemocratic bureaucratic machine.

One thing many in Brussels now agree on is that something has to change in the way the European Union works. But deciding what that is, exactly, will not happen swiftly, Frans Timmermans, the Dutch first vice president of the bloc’s executive body, the European Commission, cautioned Monday in a post on Facebook.

“The Brexit vote is not an isolated incident and is not just about Europe,” Mr. Timmermans wrote. “It is also about a broad sentiment in Western societies that we have lost control of our destinies.”

The search for some sort of answer will start on Tuesday at a previously scheduled two-day summit meeting of European leaders. It will be the first full gathering of the leaders, including the British prime minister, David Cameron, since last Thursday’s referendum. The group will consider for the first time how to respond to the British vote.