Back in the 1200s green tea was drunken in powdered form, in bowls, wisked to a froth with a bamboo wisk. This occurred in China, Korea and Japan. At the end of the 1200s the mongols invaded, somehow wiping out this tea culture. As the mongols weren't able to successfully invade Japan, it survived there, later becoming the Japanese tea ceremony.
In Japanese and English it is called Matcha, in Korean its Malcha 말차. In Busan today, there are two places that serve green tea that way, the best of which is in Someyeon at a place called DaSoul.
SAs you can see on the map here DaSoul is located next to the McDonalds and the public library. To get there, take the street between The Burger King and the underground McDonalds heading away from the underground shopping mall street.
Open from 12 till 12 they offer a wide variety of the traditional Korean teahouse teas but are known for their Matcha or Malcha as it is called here in Korea.