“I learned so much from having to deal with stuttering,” Mr. Biden said in a 2016 speech at an American Institute for Stuttering gala. “It gave me insight into other people’s pain.” 

When Mr. Biden met a young boy named Brayden Harrington in a rope line on the 2020 campaign trail, Mr. Biden was told that the boy had a stutter and promptly invited him backstage to chat. Mr. Biden recommended that Brayden read a book by one of his favorite Irish poets, William Butler Yeats, to help him envision speech like a poem. He also showed him the notes he used for the day’s speech.

“After every couple of lines or words he would draw a line straight down, a blank space between words, and that would indicate for him to take a breath,” Brayden, 15, said in an interview. He added that when they first met, Mr. Biden “looked me right in the eyes and said, ‘Aw, man, your imperfections are your gifts.’”

As president, Mr. Biden frequently describes his stutter as part of a painful past he will not return to. “It cannot define you. It will not define you. Period,” he said, at a November campaign event in California, after he saw a person in the audience holding up a sign that said, “Thank you for having a stutter.”

Like most White House traditions, the State of the Union address takes on the personality of the man giving the speech. So do the preparations.

Most modern presidents make notes on their significant speeches. President Ronald Reagan made “hash marks” to divide his speeches into 30-second chunks. President George W. Bush, who was not known as a strong public speaker, practiced with small notecards and underlined words for emphasis. President Barack Obama worked with writers — including one he had christened with a lofty nickname, “Hemingway” — and then rewrote the entire speech in his own hand. President Donald J. Trump claimed that he wrote all of his speeches (he did not) and then scribbled edits with a Sharpie.

In the Biden White House, once a serviceable State of the Union draft has been created — after several rounds of preparation between Mr. Biden and his team — a larger group of aides is pulled into the process. Last year, a video released by the White House showed several of Mr. Biden’s closest aides, including Ms. Dunn; Kate Bedingfield, the White House communications director; and Jen Psaki, the former White House press secretary, reviewing the material and taking notes.