Menken was given an honorary Emmy in 1990 but honorary awards aren’t counted toward the EGOT.
Menken and Slater had previously been nominated for a pair of Primetime Emmys for musical work done for the series Galavant and The Neighbors.
The Emmy was the award that eluded him the longest, but now he has it, as the Daytime Emmy Awards were recently given out, and (via EW) Alan Menken and Glenn Slater won the award for Outstanding Original Song in a Children’s, Young Adult or Animated Program, for the song ‘Waiting in the Wings” from Disney’s Tangled: The Series.