Vonnegut split writers into two camps, and he was firmly in one of them. The method he used wasn't fast, and he never pretended it was.

The Direct Answer
One sentence at a time, perfected before moving on.
He called this being a basher, in his 1997 novel Timequake, and counted himself among them.

Swoopers and Bashers

In Timequake, Vonnegut described two kinds of writers. Swoopers race through a draft fast and messy, then go back and fix everything that's broken. Bashers do the opposite. They grind out one sentence at a time, getting each one exactly right before they let themselves move to the next.

He put himself squarely in the second camp. A basher reaches the end of a draft and is more or less finished, because the polishing happened along the way. It's slow, exacting work, and Vonnegut owned it.

Bashers go one sentence at a time, getting it exactly right before they go on to the next one. When they're done they're done. I am a basher.

Kurt Vonnegut, Timequake

Why the Slow Way Worked for Him

Bashing fit the rest of his life. His writing hours ran from before dawn to mid-morning, a tight window where deep focus on a single sentence was possible. He wasn't trying to hit a word count or sprint to a finish. He was trying to get each line to carry its weight, which lines up neatly with his own rule that every sentence should either reveal character or move the action.

For the rules he wrote those sentences by, and the schedule that surrounded them, see the full Kurt Vonnegut writing routine.

What You Can Steal