Caramelize the onions: Cut the onions into ¼-inch slices. Melt the vegetable oil and butter over high heat in a large sauté pan. Add the onions, reduce the heat to medium, and sauté until the onions turn translucent. Stir occasionally, reducing the heat as needed to prevent burning. Continue to cook until the onions turn a deep rich brown (without burning), about 45 minutes.

When the onions are done, place them in the bottom of a pressure cooker (if using) or a large stock pot.

Generously season the beef short ribs on all sides with salt and pepper. Using the same pan that you used for the caramelized onions, set the pan on high heat. Sear the ribs on four sides, about 2 minutes per side until nicely browned. Remove the ribs from the sauté pan to the pot with the caramelized onions.

Deglaze the sauté pan with the kimchi over high heat, rubbing the kimchi around the pan to sop up the fond from the short ribs. When the bottom of the pan is mostly clean, scrape the kimchi and juices into the pot with the onions and beef ribs.

Pour the dashi into the pot. If using a pressure cooker, cook on high for 1 hour and allow natural release. If using a conventional stock pot, bring the contents to a boil, cover, and reduce to a simmer for 2 to 2.5 hours until the short ribs are tender.

When the jjigae is almost done cooking, heat the sesame oil in a large sauté pan. Sauté the garlic until fragrant (about 30 seconds), then add the rice cakes and stir-fry until lightly browned.

Stir the rice cakes into the jjigae and simmer for about 5 minutes or until the rice cakes are tender and chewy.

Season the jjigae with salt to taste, then stir in the mirin. Serve the jjigae topped with bean sprouts, green onions, carrot, enoki mushrooms, and white and black toasted sesame seeds.