To prepare the soy and ginger glaze and marinate the duck
Combine all of the ingredients for the glaze in a sauce pot and slowly bring to a simmer. Continue to slowly cook for 15 minutes, then remove from the fire and let the ingredients steep to cool. Place the duck breasts in a shallow container, skin side up, in a single layer. Pour the cooled glaze over, cover and refrigerate. Marinate for six hours, turning them over halfway through.
Combine the vegetable oil, chili flakes, and cayenne pepper in a pot large enough to accommodate all of the remaining ingredients. Heat slowly over a low flame to infuse the chili flavor into the oil for 10 minutes. Add the remaining ingredients to the pot and slowly bring to a simmer. Remove from the fire and cover. Let stand at room temperature until cool, then strain through a fine sieve, discarding the solids. Transfer to a container with a tight fitting lid and refrigerate until ready to use.
The recipe for the ponzu sauce yields more than will be needed for this dish, but it is a great thing to have on hand for other dishes. In the restaurant we always have ponzu on hand, to dress a quick tuna tartar or even a beef carpaccio.
To assemble the dish
Ready the grill. Remove the duck breasts from their marinade and pat dry. Brush lightly with oil and grill. The duck should be grilled over coals that have been burned nice and low to avoid the flames from jumping up as the fat renders. Also brush the mushroom caps with oil, season, and grill alongside the duck breasts. Remove the duck breasts from the fire after they have reached the desired doneness. Remove the mushrooms and cut into thick strips. Add the mushroom strips to a bowl with the soba noodles, radish circles, baby arugula, and a sprinkling of sesame seeds. Dress the salad liberally with ponzu sauce and toss. Divide the soba noodle salad evenly between plates, slice the duck breasts and lay over the salads. Top each with crisp shallots and serve.