The description of the illusionist's bracers (GGR, page 178) says:
While wearing the bracers, whenever you cast a cantrip, you can use a bonus action on the same turn to cast that cantrip a second time.
The sorcerer's Twinned Spell metamagic option (PHB, page 102) says:
When you cast a spell that targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).
Can I use Twinned Spell when I cast booming blade (SCAG, page 142) as an action, and using the illusionist's bracers, twin-cast booming blade again as a bonus action (thus attacking 4 targets or the same 2 twice each)?
Best Answer
Update: the Sword Coast Adventurer's Guide errata from November 2020 changes Booming Blade's range to self, making it uneligible for twinning specifically. The following applies to Booming Blade only if you don't use that errata.
The only general limitation on metamagic, as stated in PHB, is:
But there is no limitation on how many times you can use metamagic per turn. And the castings with these bracers are explicitly written to be two castings (first as an action, second as a bonus action), as quoted in your question.
So yes, you can twin the spellcastings as you describe, an action and a bonus action, in the same turn. You do have to spend sorcery points for both instances of Twinned Spell (2 sorcery points total, as both spells you're twinning are cantrips).