Bring wild rice and its 3 cups water to a boil in a saucepan. Cover and simmer on low for 45 to 55 minutes until tender with some grains split, then drain excess water.
Meanwhile, heat the oven to 425°F. Toss squash with 2 teaspoons oil on a rimmed sheet and roast for 25 to 30 minutes, turning once, until tender and browned at the edges.
Heat remaining oil in a 6-quart soup pot and cook onion for 5 minutes. Add garlic and sage for 30 seconds, then broth, salt, pepper and roasted squash; simmer for 5 minutes.
Blend the squash mixture smooth with an immersion blender, keeping the blade submerged. Stir in cooked wild rice and roast turkey and simmer gently for 5 to 8 minutes until the turkey reaches 165°F.
Add half-and-half and warm gently for 2 minutes without a hard boil. Add reserved broth in small amounts if needed to keep the soup pourable and serve.