Preheat the oven to 400℉/204℃. Line a baking sheet with parchment paper or a silicone mat.
Prepare the Kuri squash for roasting. Wash and scrub the outside to remove dirt, then pat dry. Use a large, sharp knife to slice off the stem and a little from the bottom so that it stands up straight on a cutting board. Then, guide your knife through the center starting at the stem end to cut it in half lengthwise. Push the base of your hand on the knife to safely guide it through as needed. Scoop out the seeds from each half and discard.Rub the cut side of each half with about a tablespoon of olive oil. Sprinkle with salt and pepper. Place cut side down on the prepared baking sheet. Bake for 40 to 50 minutes (and up to an hour if needed) until the flesh is tender and you can easily pierce through the tough skin with a fork. Flip the halves over so they are cut side up and can cool off while you prep other ingredients. You can transfer them to a plate and put it in the fridge to accelerate the cooling if needed. While the squash is cooling, prep the apples. Add butter and diced apples to a medium skillet over medium heat. Cook, stirring occasionally, for 7 to 10 minutes until the apples are starting to soften. Stir in the brown sugar and cinnamon and remove from heat.
If the squash is now cool enough to be handled, start the soup. In a large soup pot, warm the other tablespoon of olive oil. Add the leeks and cook for 3 to 4 minutes until softened. Stir in the garlic, ginger, thyme, salt, and pepper. Cook for 1 more minute. Scoop the flesh out of the skin of the squash and put it into the pot. Add the broth, increase heat to bring to a boil, then reduce heat to maintain a simmer for 5 to 10 minutes.
While the soup simmers, prepare the last topping of blue cheese drizzle. In a small saucepan, warm the butter and cook the minced garlic for a minute until fragrant. Add the blue cheese and half and half. Cook over medium-low heat, stirring frequently, until thickened. This usually takes about 5 to 10 minutes. Remove from heat.
When the soup has finished simmering, remove from heat. Blend it right in the pot using a handheld/immersion blender. See notes for upright blender instructions.
Serve the soup in bowls with the caramelized apples, creamy blue cheese drizzle, and chopped walnuts. Enjoy!
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