 If you liked
Dae Jang Geum: Jang Geum's Dream
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 ...then you might like
Seirei no Moribito
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Very different stories but both have historical East Asian settings and a young/teenage protagonist who goes on a journey filled with action, personal growth, and form friendships with people who are much older than them. Seirei No Moribito is more fantasy/supernatural, while Dae Jang Geum is focused on cooking the finest Korean cuisine - but both could have been set in an overlapping universe.