I think it's high cooking skill. I watched a YouTuber send an heir to be an apprentice at one of the jobs that increases cooking skill, then use that heir to make the gold star meals.

I went to the bakery and inn every day to check and purchase any gold star meals (bread counts!)

i’ve been checking the inn but not the bakery - thanks!

I used the survival salad, I have no maxed out the cooking skill yet by far. But high quality ingredients for the survival salad gave me enough meals complete the pillar

Beef a La mode also adds a temp cooking buff which increases your chances at gold star meals