Start the soup: Drizzle olive oil into large roasting pan. Add onion, carrots, garlic cloves, and a big pinch of salt; toss until vegetables are coated in oil. Place squash halves onto the chopped vegetables with the cut sides facing up. Sprinkle squash with additional salt.

Roast in the preheated oven until squash is tender and easily pierced with the tip of a knife, about 1 hour. Remove from the oven and allow vegetables to cool a bit.

While the vegetables are cooling, make the brown butter: Melt butter in a saucepan over medium heat. When butter foams and starts to turn golden brown, reduce the heat to low. When butter turns from golden to a light, nutty brown, remove from the heat and immediately stir in sage leaves.

Finish the soup: Scoop out squash flesh and place into a large stockpot with remaining roasted vegetables. Stir in chicken broth, brown butter mixture, and maple syrup. Place the pot over high heat and bring to a simmer. Reduce the heat to low and slowly simmer to blend flavors, 45 to 60 minutes. Remove from the heat.

Blend with an immersion blender until very smooth, 3 to 4 minutes. Pass mixture through a strainer to remove any remaining fibrous bits.

If soup seems too thick, add a few tablespoons of water or chicken broth. Stir in cider vinegar, a pinch of salt if needed, and cayenne pepper. Garnish each serving with a dollop of crème fraîche and chopped chives.