Hi! I'm not sure what distinction you're making between "the first sound of the letter" and "the whole letter", unless by the former you mean that the pronunciation of it begins with a vowel sound, and by the latter, you mean the fact that it's a consonant by definition.
The only reason we have both "a" and "an" in English is for ease of pronunciation, unlike other languages where alternative articles actually indicate a difference. So the rule is: always choose the one that produces something easier to say.
Is it easier to say "A LLC" or "An LLC"? Since "L" is normally pronounced "ell", it has to be the later!
Don't be fooled by the vowel vs consonant thing that you learnt at school! It's the sound that matters. Compare "a union" and "an umbrella", or "an elephant" and "a euro" - it's not the letter as such, but how you say it that determines whether you say "a" or "an" before (hence "a man" but "an old man").
"He who dares not offend cannot be honest" Thomas Paine