Heat 2 tablespoons oil and 2 tablespoons butter in a large saute pan over medium heat. Add onions and saute until softened, about 8-10 minutes.

Season onions with salt and sugar, stirring to combine. Continue to saute until onions are a deep brown color and caramelized, stirring occasionally. If the pan becomes too dry, add a splash of water.

Add garlic and saute for another minute.

Whisk together Shaoxing wine and white miso until miso is dissolved. Pour miso mixture into pan and bring to a simmer. Continue for simmer 1 minute or until almost all of the liquid has evaporated. Set aside.

Cook noodles according to directions on package. Reserve 1 cup of the cooking liquid and drain noodles.

Add noodles to caramelized onions along with reserved cooking liquid, and remaining 2 tablespoons butter. Stir until butter has melted. Season with soy sauce and black pepper.

Portion noodles into bowls and top with fried eggs, chopped scallions, furikake, and chili oil. Serve immediately.