Step 1

Cook garlic and ¼ cup oil in a medium pot over medium heat, stirring often, until garlic is beginning to turn golden, about 3 minutes. Stir in sesame seeds and cook, stirring occasionally, until garlic is golden brown and crisp, about 1 minute. Transfer mixture to a small bowl and stir in gochugaru; season with salt. Set garlic oil aside. Wipe out pot and set aside.

Step 2

Trim dark green parts from scallions and thinly slice; set aside for serving. Coarsely chop white and pale green parts. Heat remaining 2 Tbsp oil in reserved pot over medium-high. Cook chopped scallions and ginger, stirring often, until scallions are charred in spots, about 4 minutes. Add tomato paste and cook, stirring occasionally, until it begins to stick to the bottom of pot and darkens slightly, about 2 minutes. Add mushrooms and kombu, then stir in 5 cups cold water. Bring to a boil, then remove from heat and let sit until mushrooms soften, about 10 minutes. Remove and discard kombu.

Step 3

Using a slotted spoon, transfer solids to a blender. Add a ladleful or 2 of broth to blender and purée until smooth. Stir purée back into broth in pot and bring to a simmer over medium heat. Add butter a piece at a time, whisking to combine after each addition before adding more. Stir in soy sauce; season with salt. Reduce heat to low and keep warm until ready to serve.

Step 4

Meanwhile, bring a large pot of water to a boil. Add bok choy and cook until bright green and tender, about 2 minutes. Using a slotted spoon, transfer bok choy to a plate. Return water to a boil and cook noodles according to package directions. Drain and divide among bowls.

Step 5

To serve, ladle broth over noodles, then top with bok choy and reserved garlic oil. Top with eggs, nori, and cilantro if desired.