Preheat & Prep: Preheat oven to 325°F. Shred cheeses and divide into three 1-cup portions (for sauce, middle layer, and topping).
Cook Pasta: Bring a large pot of salted water to a boil. Add elbow pasta and cook 1 minute less than package directions for al dente. Drain, toss with a splash of oil to prevent sticking, and set aside.
Make the Roux & Sauce: Melt butter in a large saucepan over medium heat. Whisk in flour and cook for 2 minutes. Reduce heat to low. Gradually whisk in half-and-half and milk in small additions, whisking constantly until smooth and lump-free (about 5 minutes).
Combine: Remove saucepan from heat. Stir in 1 cup of shredded cheese until fully melted. Fold in the cooked pasta, half a package of chopped J-Basket Kimchee, and the other seasonings.
Layer & Bake: Pour half of the pasta mixture into the 9x9 baking dish. Scatter 1 cup of cheese across the center, then spread the remaining pasta mixture on top. Finish with the final 1 cup of cheese and the remaining 1/2 package of J-Basket Kimchee. Bake for 15 minutes, or until the cheese is melted and golden brown on top.
Garnish & Serve: Remove from oven, sprinkle generously with J-Basket Crunchy Furikake and chopped green onions, and serve warm.