The stoic. The quiet one. Violence comes naturally to them, and they do it all with a straight, unbothered poker-face. Not one for idle talk, they rarely speak, which means that when they do, their words carry extra weight. Their silence could be the result of being an unfeeling psychopath, or just damaged and shy. Anton Chigurh, the antagonist of No Country For Old Men (2007), personifies the latter. A remorseless, unstoppable force of violence, Anton made an everlasting impression on everyone that saw the film. While his appearance, with the strange mop-top haircut, may not signal "coolness" at first glance, it is his demeanor that really does it.
Flipping a coin to decide the fates of his victims, (Harvey Dent called, he wants his shtick back) Anton displays a puzzling dedication to chance, which is contradictory to his otherwise practical, all-business demeanor. This contradiction only serves to make him more mysterious, and therefore cool. His weaponry is pretty epic too. A captive bolt stunner, a brutal tool designed for stunning animals, repurposed as a nifty little killing tool. It's unconventional and eccentric, but gets the job done, which is somehow cool. Not to mention his Remington shotgun, with its hefty suppressor, putting the "silent" in silent badass.
Now while Anton may be a homicidal psychopathic villain, killing innocents left and right every time he needs a new vehicle, strangling deputies with handcuffs, murdering people simply for annoying him on the authority of a coin toss, nobody can deny he is cool as hell. Perhaps it is unrelenting violence that makes him cool, or his unsettlingly eccentric behavior, or just his unstoppability as an evil force. Either way, Anton has to be the quintessential silent badass.
Another character that's a little more silent and a little less badass is the Driver from Drive (2011). The protagonist of Nicolas Wending Refn's cult classic that spawned an aesthetic drooled over by film-bros for years to come (I do not claim to not have been one of them) is often cited as a "silent badass" and although he definitely is silent and arguably a badass, I don't think he personifies the silent badass nearly as much as Anton. His silence seems to be more as a result of having not much to say, and having not much going on in his head. Don't get me wrong, I love the movie and the character, but he's a blank slate of a character, and his silence often makes him come off as awkward instead of cool.
You need the unspoken power to speak for you. The strong-yet-silent routine does not work if the character is merely fronting as a badass
Nevertheless, the Driver does shine in his combat sequences, which is badass, and he is silent, ergo, he's a silent badass and therefore cool. If I sound dejected, it's because on rewatches of the film I've come to see him as more of a weird, violent idiot, but I digress. See also: Ryan Gosling's other character directed by Refn, Julius from Only God Forgives (2003). He's practically the same character - Silent Ryan Gosling badass - with more backstory and considerably bigger mommy issues.
I'd be remiss if I didn't mention Hugo Stiglitz, the stone-faced nazi killer from Inglourious Basterds (2009) as another cool silent badass. Now undoubtedly, the whole squad is cool as a cucumber, but Hugo, the former-SS officer turned SS officer former(izer), is the "silent badass" cool. He keeps a stone-face as he chokes, stabs or whacks Nazis, and equally stone-faced as he rots in jail for it and is then rescued in a hail of gunfire. In my mind, I've always likened him Wolfenstien's to B.J. Blaskowicz, another silent badass "Aryan" Nazi-killer, from the video-game world.
A few more things about this archetype: merely being silent and violent do not qualify you to be a silent badass. You might have the demeanor, but you need the unspoken power to speak for you. The strong-yet-silent routine does not work if the character is merely fronting as a badass, while actually being pretty stoppable.
An example of this distinction could be Gaear Grimsrud from Fargo (1996). He sure may say little and kill without remorse, but behind his silence is a lack of brains or brawn or both. He acts recklessly and gets defeated by the actual (mainly) silent badass of that film, Marge Gunderson, a heavily pregnant, amazingly midwestern policewoman. She outwits the bumbling bad guys through sheer smarts, while speaking mainly in one-word sentences and being unbelievably sweet, earning her status as a shining silent badass.