Raw burdock is tough and fibrous, so be sure to use a sharp knife to slice it. The lemon in the cooking water keeps the burdock white.
Preheat the oven to 400°. Spread the walnuts in a pie pan and toast for about 6 minutes, or until fragrant.
Squeeze the lemon into a medium saucepan and add the water, 1 tablespoon of the sugar and a pinch of salt. Peel the burdock under cool running water. Cut it into 1/4 -inch slices on a sharp diagonal, adding the slices to the lemon water as you work. Bring the water to a boil and cook the burdock over moderately high heat until al dente, about 30 minutes. Drain and let cool slightly, then stack the slices and cut them into thin matchsticks.
Return the burdock to the saucepan. Add the remaining 2 teaspoons of sugar, 1 teaspoon of the rice-wine vinegar and 1 teaspoon of the miso. Cook over moderately high heat, stirring, until glossy, about 2 minutes. Transfer the burdock to a plate and let cool.
In a small bowl, whisk the remaining 2 tablespoons of rice-wine vinegar with the remaining 2 teaspoons of miso until dissolved. Whisk in the oil and season with salt and pepper.
In a large salad bowl, toss the lettuces, walnuts and burdock. Add the dressing, toss again and serve immediately.
Make Ahead: This recipe can be prepared through Step 3 up to 1 day ahead.