In a pot over medium heat, bring the ¾ cup water to a boil. Remove from heat and transfer to a bowl. Add brown sugar and stir until dissolved. Allow to slightly cool.
In a big bowl, place flour. Make a well in the center and pour half of the sugar water. With a spatula, stir to moisten flour. Add the rest of the water and using hands, continue to combine until dough forms. To check for the right consistency, pinch a small piece of dough and flatten with your fingers. It the dough breaks and falls apart, add more water in 1 tablespoon increments as needed. The dough should feel wet but not overly sticky.
Divide the dough into uniform portions. Using palm of hands, roll each portion into 2-inch size balls. Continue to roll until ball is smooth and without cracks. Place in a single layer on a platter and repeat with remaining dough. Cover with plastic film until ready to use.
Using fingers, flatten each dough ball into a sphere of about ¼ inch thick and 4-inch diameter. Place one tablespoon mung bean filling in the center and gather edges of dough around filling. Pinch dough together to completely seal and using palm of hands, roll repeatedly into a smooth ball with no cracks or openings. Gently dip balls in a bowl of water until just moistened (you can also use a spray bottle filled with water to lightly spray balls). Gently roll moistened balls in sesame seeds until fully coated, pressing lightly to make sure seeds adhere well to dough. Repeat with remaining dough and filling.
In a pot over medium heat, heat about 3-inch deep oil to about 350 F. Make sure oil is deep enough to fully submerge balls. Gently lower balls into oil and deep-fry for about 6 to 7 minutes or until golden brown and begins to float to top. Do not overcrowd pot. Leave enough room to be able to regularly turn buchi during frying to ensure even browning. Remove from pot and drain on a wire rack set over a baking sheet.
For the Mung Bean Filling
Rinse beans with cold water. In a pot over medium heat, combine beans and about 2 cups water. Bring to a boil. Lower heat, cover and cook for about 20 to 30 minutes or until tender. Remove from heat. In a blender or a mortar and pestle, process beans until smooth. Add sugar and stir until combined.
In a non-stick pan over medium-low heat, add mung bean mixture. Cook, stirring regularly and mashing with back of spoon, until liquid is absorbed and mixture turns into a thick paste. Remove from heat and allow to cool.