I have a question about the rules around gaining Fame and how it interacts with other cards.

Do I gain fame 1 at a time, or do I gain it "in bulk" all at once?

The specific interaction is between gaining Fame and a mod called Chromium Hull, which says "When you gain fame, gain 1 additional fame..." So, if I do a job or deliver cargo that gives me 2 fame, do I gain the fame 1 at a time (and thus end up gaining 4 fame total) or do I gain the 2 fame from the job, and then 1 additional fame (and thus end up gaining 3 fame total)?

I'm going to lean towards the latter. When you "gain 2 fame" you gain 2 fame; not gain 1 fame then 1 fame.

I tend to agree, but coming from X Wing, where damage is suffered one at a time, I just wanted to make sure. I'll play it the way you're suggesting unless it's confirmed otherwise.

It is the latter. It does not state that you should gain twice the fame that you would otherwise gain. It allows you to gain 1 additional fame whenever you receive an amount of fame. Having played multiple FFG games, if they intended for a 2 fame reward to be received as 2 singles then they would word the cards in the manner of "receive 1 fame, then receive an additional 1 fame" or "Repeat this effect if you are not the most famous player".

This does not double up with cards such as the cloak which considers you to have extra fame while in good standing with a number of factions, but does stack with other cards which grant additional fame when any fame is received.