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 peel eggs, and set aside.
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 small saucepan; cook over low, stirring occasionally, until hot, about 8 minutes. Add noodles to boiling water, and cook according to package directions for al dente. Drain noodles, and divide evenly among 4 bowls.
Divide butter evenly among bowls; pour hot soy sauce mixture evenly over bowls. Halve eggs lengthwise, and top each bowl with 2 egg halves. Sprinkle evenly with green onions, shredded carrots, thinly sliced red pepper, and edamame (optional). Scatter a generous amount of black pepper over noodles. Drizzle evenly with sesame oil, and garnish with sea salt.