Stuttering Into Eloquence

You have a character who stutters — badly. But then, at some key moment, his stutter gives way to a speech delivered with surprisingly smooth and eloquent diction. Stuttering Into Eloquence is commonly used to underline the point that what he is saying is dead serious.

Examples of Stuttering Into Eloquence include:

Fan Works

Up until her early twenties, Kaitlyn Hutchins of Undocumented Features suffered from a severe stutter. It vanished when she sang and when she was in the company of people to whom she was particularly close. And, relevant to this trope, it also vanished when she went full-bore samurai on someone.

Film

Shakespeare in Love, where Wabash can barely speak during the film but (after a brief false start) delivers the prologue to the play perfectly.
Subverted in Pan's Labyrinth: The Big Bad promises to release his stuttering prisoner if he can speak without a stutter. The prisoner tries and fails, so gets tortured and executed.
Michael Palin's character K-k-k-Ken in A Fish Called Wanda has a terrible stutter throughout the movie. It vanishes when he gets his revenge on Otto, his major adversary, at the end of the movie. Final credits reveal that he now works as an MC at Sea World. Also, "Cathcart Towers Hotel!"
- It also goes away in one scene where Jamie Lee Curtis' character becomes frustrated with his inability to tell her what she needs to know, and she kisses him.
The Waterboy has Bobby stop his stuttering (and add a bit of bass in his voice) when he calls out his overprotective mother over her treatment of him.
This is the entire plot of The King's Speech. Unlike most of the other examples, King George VI had to work hard to overcome his stutter, but he certainly came though when it counted. Parts of it are played for laughs when we learn that he doesn't stammer when he's shouting, swearing or singing.
Aaron Stampler from Primal Fear has a severe stutter except for when he changes to the much more aggressive "Roy" .
One of the characters in Once Upon a Time in China has a stutter based on the fact that he's not a native Chinese speaker. Near the end of the film, he suddenly breaks out the entire plot rapidly to the hero, initiating the final battle. When asked how the heck he did this, he revealed that he'd been mentally rehearsing the speech for an hour.

Literature

Brandon Mull went to town with this on Gavin in Fablehaven, who was actually very eloquent and great at dragon taming. Unfortunately, to Kendra's dismay, Gavin was really the evil dragon Navarog in human form and the stutter was simply to make him appear vulnerable. He gets a Karmic Death pretty quickly though.
Used for The Reveal in Harry Potter and the Philosopher's Stone. The villain was p-p-p-poor s-s-stuttering Professor Quirrell all along.
Bill from Stephen King's IT.
In Isaac Asimov's short story "Galley Slave", sociology professor Simon Ninheimer stuttered (or something) almost all the way through to the end, where he gives a eloquent speech of the evils of ~~spell checkers~~ artificially intelligent proofreaders. What Do You Mean It's Not Awesome? for someone I'm sure.
Was the basis of one of P. G. Wodehouse's "Mulliner" stories, "The Truth About George", later adapted for television on The BBC Wodehouse Playhouse.
The title character of I, Claudius eventually trains himself so that he largely loses his stutter. What makes it fit this well is that he had achieved this well before most people thought he had- and so right before he becomes Emperor, he's still poor C-C-C-Claudius, but after that, he drops the Obfuscating Stupidity.
Simon has a stutter throughout most of Equal Rites, but loses it after a run-in with some Cosmic Horrors.
Billy in One Flew Over the Cuckoo's Nest has a brief moment of bravery, during which he stops stuttering.

Live-Action TV

Reg Barclay in Star Trek: The Next Generation and Star Trek: Voyager had a few scenes like this. It was usually evidence that he was really certain about what he was saying. Although sometimes it was evidence he'd been hit by the Negative Space Wedgie and was Not Himself.
A character on Joan of Arcadia had a terrible stuttering problem, made worse by the fact that he was on the Debate Team. So how did he find his voice? Joan discovers that he has an exceptional talent for finding/gathering evidence and arranging excellent presentations.