Booming Blade:
You brandish the weapon used in the spell’s casting and make a melee attack with it against one creature within 5 feet of you.
Extra attack:
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
I think we can agree that Booming Blade is casting a spell (a cantrip). Obviously I can’t use Booming Blade twice.
But does the fact that it’s a spell that makes a melee weapon attack count as “taking the attack action”?
In other words - if I cast Booming Blade, am I locked out of Extra Attack because I took a “Cast a Spell” action instead of an “Attack” action? Or do I still get an extra attack because casting the spell made me take the Attack action?
As a caveat to what has been stated by others, if you’re a bladesinger wizard, the Extra Attack there does allow for it to be used with Extra Attack since Bladesinger allows one of the attacks to be replaced with a cantrip cast.
And Eldritch Knight has War Magic, which does not quite do the same thing, but still makes casting Booming Blade almost always better than the Attack action from levels 7-11 (it’s not generally a BA heavy subclass, but occasionally you’ll want to use something like Second Wind.) Even after level 11, the bonus damage puts the DPR higher than 3-4 attacks, so the choice is between more damage upfront vs. movement deterrent with a higher damage punishment.