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1
Combine 1 1/2 quarts water, kombu, and bonito flakes in a large saucepan and bring to a boil over high heat. Reduce to a bare simmer and cook for 5 minutes. Remove from heat, let cool for 5 minutes, then strain through a fine-mesh strainer. Discard solids and set dashi aside.
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2
In a large Dutch oven or soup pot, heat 1 tablespoon oil over medium-high heat until shimmering. Add leek, carrot, garlic, and sliced ginger. Cook, stirring, until vegetables are glistening and just starting to turn tender, about 4 minutes.
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3
Add squash and pour just enough dashi on top to cover vegetables. Bring to a simmer and cook until vegetables are fully tender, about 30 minutes. Using a standing blender or immersion blender, and working in batches if necessary, blend soup until very smooth. Blend in miso and lemon juice.
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4
Return soup to pot and thin with enough water to reach a pourable, silky-smooth consistency. Season with salt, add sugar to taste, and keep warm.
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5
Meanwhile, fill a medium bowl with 2 cups water and 1 teaspoon kosher salt. Add diced apple and let soak for 10 minutes. Drain apple well, then return to bowl. Toss with grated ginger, scallions, toasted sesame seeds, sesame oil, rice vinegar, and remaining 1 teaspoon vegetable oil. Season with salt, if needed.
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6
To serve, ladle hot soup into bowls and top with the apple-scallion salad. Garnish with shichimi togarashi, if desired.
