BRUSSELS — On the menu was a starter of green asparagus and smoked salmon, with a main course of European discord.

After a two-day meeting in Brussels, European Union leaders said on Friday that they had failed to agree on who will lead the bloc’s key institutions — first and foremost the European Commission, the administrative branch — for the next five years.

In typical European Union style, as the leaders squabbled over dinner on Thursday in a room with jammed cellphone signal and few aides, they even disagreed on the process they would follow to reach a decision.

They said they would meet again on June 30 to give it another go.

The tussle over this esoteric but important choice of personnel to lead the institutions that make the bloc work — including the critically important European Central Bank, which sets monetary policy for the 19 member states that use the euro — is emblematic of the political disharmony at the heart of the world’s richest group of nations.