Preheat oven to 375°. Cut pumpkin and squash in half lengthwise. Scoop out seeds and any stringy parts. Put flesh side up in a large roasting pan with 1 cup broth. Cover pan with foil and bake until vegetables are tender when pierced with a fork, about 1 hour.
Meanwhile, melt 3 tbsp. butter in a large pot over medium heat. Add onions and 1 tsp. salt. Cook, stirring occasionally, until onions are soft and start to look creamy, about 5 minutes. Reduce heat to low or medium-low and cook onions, stirring every few minutes, until they turn a caramel color and become quite sweet, about 30 minutes. Set aside.
When pumpkin and squash are tender, scoop out flesh and set aside; discard skins. Reserve any liquid in bottom of pan.
Return pot with onions to medium-high heat. Add garlic and 2 tbsp. fresh ginger. Cook, stirring, until fragrant, about 2 minutes. Add ground ginger, nutmeg, cloves, and cardamom. Cook, stirring, 1 minute. Add remaining 7 cups broth, the carrots, cooked pumpkin and squash, and reserved liquid from roasting pan. Bring to a boil, then reduce heat and simmer until carrots are tender, about 15 minutes.
Whirl vegetables in a blender (in batches) until completely smooth. (For silky-smooth soup, you can pour the puréed soup through a strainer.) Return to pot and stir in brown sugar. Season with salt to taste. Keep warm over low heat.
Put a small bowl or measuring cup next to the stove. Melt remaining 4 tbsp. butter in a small frying pan over medium-high heat. Add remaining 1 tsp. fresh ginger. Cook, stirring occasionally, until butter starts to foam. Stir mixture constantly until it starts to brown. Pour mixture into waiting bowl or measuring cup. Divide soup among 8 bowls and serve hot, with a swirl of ginger browned butter in each serving.
Note: Nutritional analysis is per serving.