In this au, Gaster isn't killed in the core accident, and so he's present for the major tragedies that would normally follow his death. And subsequently is able to fix some of the problems! One being Asriel's death and transformation into Flowey.

Gaster, who was working on making a monster from almost scratch (his soul was too weak to form another soul but he wanted a kid and yadda yadda), successfully created Sans. After a bit of studying Sans' growth, Gaster was practically able to revive Asriel. Asriel is older than Sans by a few years as his 'death' made him miss a few.

And during this process, Gaster also managed to rizz up Asgore by complete accident. They'd always been close, even before the war, but Asgore was to be king, and he was already betroathed to Tori at birth, so Gaster was content being by Asgore's side as a friend. Buttttttt Toriel leaving for the ruins in her grief left the door wide-open for Asgore to finally make a move on Gaster. They never officially marry, but literally everyone outside the ruins is aware that the two are together.

Asriel remembers very little of his death, but he remembers Chara, and he remembers the surface. Asgore and Gaster don't hide the fact that Asriel died from him, and one day Azzy got up the courage to ask Gaster if he could bring back Chara too. Gaster hadn't even thought about it, but he figured that without Chara's body he wouldn't be able to revive them in any meaningful way. Azzy doesn't give up hope, but he doesn't bother Gaster about it anymore.

Azzy is a gentle soul, shy and playful. He's not used to being an older sibling, and he misses his mom, but he likes his new family too. He often trips and falls, resulting in the bandaid scattered on his body, and he loves to farm just like Asgore, so he usually wears overalls.

Sans is Gaster's oldest son by magic-line. He's punny, a prankster, and the more confident between himself and Asriel. If they're out of their element, Sans is the one to forge through the path and laugh while doing it! Gaster can't get him to wear shoes ("You and Dad and Azzy don't, so why should I?" Is usually his rebuttle. No one can really argue it.) So he's usually seen only wearing socks.

Papyrus (who is only a few scribbles in Gaster's arms rn) is the youngest and doesn't really get the same chances as Sans and Azzy do as older kids, but he'll grow to be the most rambunctious of all of them. I think Papy is self-conscious when he's young (what if he's not good enough? Sans is so good at science stuff like Papa, and Asriel is a great prince like Dad, so where does that leave him?) But he ends up training in combat with Asriel before meeting Undyne and deciding he wants to be part of the royal guard! (That way he can protect his 1 hp brother and his Princely brother!) Undyne and him are quick friends.

Actually, since I'm on a role: Asriel keeps studying and learning from his dad on how to be a prince. Honestly, he likes what Sans and his Pops are up to a lot more, but he doesn't mind being Future King. (He sneaks into the lab to watch the two do their studies whenever he can.) Meanwhile Sans works with Gaster in the labs! He's a fast learner and loves the work, but everyone can see just how much more natural he is with social science. He understands the political theory Asriel tells him and Sans can dissect exactly wtf it means and why Asriel has to react the way he does. Sans still ends up being the Judge, but largely keeps his focuses to the Lab with his pops abd Alphys. And Papy is certainly Great. He ends up as co-captain of the Royal Guard. He's not quite as personable as Undyne, but he sure is committed, and has direct ties to the royal family. More often than not, when he's not doing his rounds or helping with training, he's Azzy's personal guard.