Preheat the oven to 400°F and line a baking sheet with parchment paper.
Cut the pumpkin in half and scoop out the seeds. Drizzle with olive oil and pinches of salt and pepper and place on a baking sheet, cut-side down. Roast for 50 to 60 minutes, until very soft. When cool to the touch, peel away the skin and measure 4½ packed cups of the flesh.
Heat the 2 tablespoons of oil in a large pot over medium heat. Add the onion, salt, and several grinds of fresh pepper and sauté until soft, 5 to 8 minutes. Add the garlic, ginger, cumin, coriander, turmeric, and cardamom and stir for 30 seconds, until the spices are aromatic. Add the cooked squash, coconut milk, and broth and stir to combine. Simmer for 20 minutes, then stir in the vinegar.
Transfer to a blender, working in batches if necessary. Blend until smooth. Kabocha squash can vary in water content, so if your soup is too thick add up to 1½ cups water (½ cup at a time) to reach your desired consistency. Garnish with additional coconut milk, fresh black pepper, and microgreens, if desired.