Step 1: Fill a small saucepan with water; bring to a boil over high. Lower eggs into water, and cook 6 minutes. Transfer eggs to a large bowl of ice water. Let cool 5 minutes. Drain and carefully peel eggs (the yolks will be slightly runny inside), and set aside.
Step 2: Bring a large saucepan of salted water to a boil over high. While water comes to a boil, stir together vegetable stock, tamari, and mirin in a separate, small saucepan; cook mixture over low heat, stirring occasionally, until hot, about 8 minutes.
Step 3: Add noodles to boiling water, and cook according to package directions for al dente. Drain noodles, and divide evenly among 4 bowls.
Step 4: Divide butter evenly among bowls; pour hot soy sauce mixture evenly over bowls.
Step 5: Halve eggs lengthwise, and top each bowl with 2 egg halves. Sprinkle evenly with scallions, and grind a generous amount of black pepper over noodles. Drizzle evenly with sesame oil, and garnish with sea salt.
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