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an, because the spoken sound of "LLM" starts with a vowel, "el". So, pronunciation of first letter of initialism, when you're spelling the initialism. I think.

an LLM, an MP, an FBI agent, a CIA agent, a NASA engineer, an NSA mechanic, an 8 year old car, a 10 year old dog, et cetera – it's the sound of how you said those letters, always the sound.

Also [a history vs. an history][1].

[1]: https://www.reddit.com/r/grammar/comments/1ssrby/an_history_...