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?
Yes. Extra Attack specifies that you must take the Attack action. With Booming Blade, you’re still taking the Cast a Spell action, the singular melee weapon attack is simply how the spell is cast, not an entire Attack action.
This is the same for all spells that require you to make a weapon attack as part of casting it. The only spells that kind of break this rule are the various “Smite” spells, but they’re worded like “When you next hit a creature with a melee weapon attack…”, and generally use a bonus action rather than an action, so you can still use your action for the Attack action.
Also the bladesingers extra attack feature at level 6 is kind of designed to synergise with cantrips like booming blade and green flame blade.
The OneDND version of EK has this at level 7 now as well