With the peel on, cut both ends of the banana. Peel the banana and cut lengthwise. Set aside. Put 1 cup of the brown sugar on a plate and set aside.
Take 1 sheet of the lumpia wrapper and put the banana diagonal along the corner of the lumpia wrapper. Place a couple strips of the jackfruit on top of the banana. Roll the banana in the wrapper to the halfway point and fold the ends in. Finish rolling the banana in the wrapper. See pictures in the Tips section.
Using a pastry brush, seal the turon with the egg white by brushing the egg white over the wrapper.
Brush all sides of the turon with melted butter and roll in the plate of brown sugar. Repeat these steps for all sliced bananas.
In a large frying pan, fill the pan halfway with oil and heat. Once the oil is hot, put about half of the remaining 1 cup of brown sugar in the oil. Once the sugar looks melted and is floating, place half of the turon in the frying pan. Note: This will depend on how large your frying pan is. You don't want to overfill the pan, so only place enough that it's not compacted. Adjust the brown sugar as necessary.
Fry the turon until golden brown and the extra brown sugar sticks to the wrapper about 2-3 minutes per side. Enjoy hot!