First, the story is based on a real historical figure, Kim Yu, a Joseon-era scholar, who wrote a book called Suunbapjang, which holds a valuable place in history of Korean Gastronomy and was classified as a national treasure in 2021. The story takes place in an era where men, or atleast noblemen of high birth, didn't cook and kitchen duties were relegated to women and lowerclass men.

Kim Yu is the second son of an honourable scholar who, along with his elder son, was framed by a rival and dismissed from court and sent home in disgrace. Now, their only hope for the revival of family honour is thrice failed Kim Yu.

On route to exam centre, Kim Yu meets Gye Am, a travelling chef, who is self-proclaimed as the best chef in Joseon and after some petty squabbles, starts working as his assitant/sous chef, learning about food and gastronomy. The rest of the movie is pretty much Kim Yu's journey as he learns about food from Gye Am, while letting go of his duties and expectations as a "nobleman scholar" and following his own path.
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SPOILER:

Towards the end, Kim Yu and Gye Am enter into a contest to become palace chefs, but Gye Am's injured arm causes him to sit back and KimYu pretty much shoulders the audition, with some help from his elder brother's hat (the best scene IMHO). Unfortunately, as a person of high birth, he can't work as a palace slave, so Gye Am alone gets the honour.

Kim Yu, meanwhile, opened his own restaurant which is open for people of all classes. He started writing his cookbook, Suunbapjang, which was eventually finished by his descendents. The book was eventually classified as natural treasure in 2021.

We DON'T KNOW whether Kim Yu and Gye Am ever met again.