1 1/2 quarts plus 2 cups water, divided, plus more as needed
1/2 ounce kombu (approximately a 4- by 6-inch piece; see notes)
1/2 ounce grated bonito flakes (about 3 cups; see notes)
1 tablespoon plus 1 teaspoon vegetable oil, divided
1 leek, white and light green parts only, diced (about 1 1/2 cups)
2 medium carrots, diced (about 1 cup)
2 medium cloves garlic, sliced
2 (1 1/2-inch) knobs ginger, 1 knob peeled and thinly sliced, 1 knob peeled and finely grated, divided
1 (2-pound) squash, such as kuri, kabocha, or butternut, peeled, seeded, and diced
2 tablespoons white or red miso paste
1 tablespoon fresh juice from 1 lemon
1 teaspoon kosher salt, plus more as needed
Pinch sugar, if needed
1 large crisp apple, such as Fuji, peeled, cored, and diced
1 large or 2 medium scallions, white and light green parts only, thinly sliced on the bias
1 teaspoon toasted sesame seeds
1/2 teaspoon toasted sesame oil
1 teaspoon rice vinegar