Get a skillet warm over a medium-high heat cooking fire or stove. Melt the butter and then place in your chopped onions. Toss the onions in the butter and heat for 3 minutes. Now toss in the sugar and salt.
Fold the onions every 3 minutes for 15 minutes, then reduce the heat to medium-low and fold every 10 minutes until they're soft and caramelized. This may take about a half hour.
While the onions are cooking, dissolve the sugar and water from the sauce ingredients by bringing them to a boil. Remove from heat, let it cool a bit, then throw in all the rest of the sauce ingredients and stir.
In another skillet, pour the mirin and ramen broth in over a high heat cooking fire or stove. After 3 minutes add in your sauce and let simmer over medium heat for 5 minutes.
Stir the onions into the sauce, reduce the heat to low and cover.
Take your now empty onion skillet and heat the vegetable oil over medium-high and then put in the rice cakes. Sizzle for 3 minutes per side until you get a crisped brown surface. Move these to a cutting board and chop into 1 inch or so segments. Toss the segments in your sauce.
Serve, garnishing with sesame seeds and green onions. This will grant your Dungeons and Dragons party a +1 Constitution stat boost in their next session!