With a sharp knife, score the skin of the duck breast diagonally every 1/4 inch (6 mm), being careful not to slice all the way to the meat, then repeat in the opposite direction to create a crosshatch pattern.
Season the duck breast on both sides with 1 teaspoon salt and let sit at room temperature for 20 minutes.
Place a medium skillet over medium heat and add 1/4 cup water. Place the duck in the pan, skin side down, and cook until the water has completely evaporated, 6 to 10 minutes. Continue to cook until the skin is browned and crisp, 10 to 15 minutes more. Flip the duck breast so it’s skin side up and cook to your desired doneness, about 3 minutes for medium-rare or 5 minutes for medium-well. Transfer the breast to a cutting board.
Drain the fat from the skillet and return it to medium heat. Add the scallions and cook until lightly browned and wilted, 2 to 5 minutes.
Meanwhile, in a medium pot over high heat, bring 6 cups water to a boil and season liberally with salt.
In a separate saucepan over high heat, combine the dashi, soy sauce, mirin, and 1 teaspoon salt and bring to a simmer.
Cook the soba in the boiling water according to the package instructions, then drain and divide between 2 bowls.
Thinly slice the duck on a diagonal. Ladle the broth over the soba and top with the sliced duck and browned scallions. Add a spoonful of spinach and serve immediately.