To make the pickling solution, place the water, rice vinegar, sugar, and salt in a saucepan, and bring to a boil. In a mason jar or airtight container, place the slices of green papaya and the baby cucumbers, then pour in the hot pickling solution. Let it cool to room temperature, and then keep it in the refrigerator. Let it pickle for at least 2 days, or for longer, up to two months to really let the flavour seep in.

Toast the white sesame seeds in a pan over medium heat until golden brown. Then, using a small blender, or a mortar and pestle, grind the sesame seeds until roughly pulverised. Then, pour them into a bowl, and add to this the Japanese mayo, rice vinegar, soy sauce, sesame oil, brown sugar, and salt.

Slice the brioche bun in half, horizontally. Then, spread a teaspoon of butter onto each side of the bun. Place the buns in a cold pan, buttered side down. Then, place the pan over medium heat, and cook it for 4-5 minutes, until nicely browned.

Once the burger buns are browned, brush the sesame dressing onto the buns. Then, stack the bakkwa and pickles on the bottom bun, and cover with the top bun. Bak-kwa burger, done.