In a bowl, whisk the peanut butter, soy sauce, rice vinegar, 2 tablespoons sesame oil, and 2 teaspoons chili oil until smooth and creamy. Taste and adjust ingredients as necessary. Set aside.
Prepare the soft-boiled eggs:
Bring a small saucepan of water to a boil. Add the egg and cook for exactly 6 minutes. Drain and rinse the egg under cold water. Carefully peel and cut in half.
Prepare the garnish:
In a bowl, combine the dark green parts of the scallions with the radishes. Drizzle with two teaspoons sesame oil and toss to coat. Season with flaky sea salt and the toasted sesame seeds and toss. Set aside.
Finish the udon noodles:
In a large bowl, combine the noodles with the white parts of the scallions, the sliced celery, and the roughly chopped celery leaves and toss to combine. Pour the spicy peanut sauce all over and toss to coat. Taste and season according to your tastes.
To Serve:
Divide the udon noodles between bowls and pile the radish garnish on top of each bowl of noodles. Serve with a soft-boiled egg, a sprinkle of toasted sesame seeds, and a drizzle more of chili oil if desired. Enjoy!