Lightly drizzle a 3-quart saucepan with oil over medium heat. Once hot, add in the whites of the green onions and sauté for one minute.
Add the garlic and ginger to the pot and drizzle with toasted sesame oil. Sauté for 30 seconds.
Add the rice and the sake and stir frequently until the wine is absorbed. Sprinkle with salt and pepper.
Stir in the kimchi (with liquid) and continue to stir frequently until the liquid is absorbed.
Next, start adding in vegetable broth, about one quarter of a cup at a time, until the rice is al dente. If needed, make sure to adjust the heat as you go (you want a simmer, but not a boil) and keep stirring frequently.
Once the rice is al dente, add in the coconut milk, agave, and the onion greens (reserve a few for garnish) and stir.
When the rice is cooked and the milk is absorbed, taste and if desired, drizzle with more agave or sprinkle in more salt or pepper. Serve into bowls or on plates and top with toasted sesame seeds and the remaining green onion.