To prepare the hot soba broth, add the dashi, kaeshi, and mirin to a saucepan and bring to a boil over high heat. Remove from the heat and set aside.
Score the duck’s skin. Season with salt on both sides, place it skin-side down on a cold dry skillet and slowly render the fat over low heat, for about 25-30 min. When the skin is golden and crispy, baste the duck with hot fat for another 5-10 minutes to cook rare (115F inner temp). Remove from heat and let rest skin-side up.
Discard the fat and return the skillet to the heat. If using, add negi and saute until it caramelizes slightly, and set aside.
Slice the duck breasts 1/8" thin and set aside, covered to prevent drying. Steps 1-4 can be made in advance and, in that case, all components of the dish should be stored refrigerated until ready to serve.
Cook the soba and drain it in a colander. If not serving immediately, cool the soba in a cold bath and reheat just before eating by dipping a strainer with noodles into boiling stock for a few seconds.
In a pot, bring the soba broth to boiling. Divide the soba noodles among warmed bowls. Arrange the duck slices and negi over the soba and watermelon slices on the side of the bowl. Pour boiling broth over the duck slices. Garnish each bowl with about 1 tbsp sliced scallions and 0.5-1 tsp wasabi. Season with shichimi togarashi to taste.