1. Melt the butter in a small pan over medium heat. Once it begins to foam, add the miso paste and stir continuously until the butter turns a nutty brown and the miso is fully incorporated. This should take about 3-4 minutes. Remove from heat and let cool slightly.
2. In a large mixing bowl, combine the cooled miso brown butter, cooked rice, green onions, beaten egg, panko breadcrumbs, soy sauce, sesame oil, and Parmesan cheese. Mix until all ingredients are well combined. Add salt and pepper to taste.
3. Shape the rice mixture into small patties, about 2 inches in diameter. If the mixture is too sticky, wet your hands slightly to make shaping easier.
4. Heat a generous amount of vegetable oil in a non-stick skillet over medium-high heat. Once the oil is hot, carefully add the rice cakes in batches. Fry each side for about 3-4 minutes or until golden brown and crispy.
5. Transfer the cooked rice cakes to a paper towel-lined plate to drain excess oil.

Remember, patience is key here — let the cakes develop that perfect golden crust for maximum flavor and texture.