Kabocha Purée with Ginger

Preheat oven to 375 °F. Line a rimmed baking sheet with parchment or foil. Season inside of squash with salt and pepper. Place squash, cut side down, on prepared sheet. Roast until tender, 1-1 1/4 hours. Let cool slightly. Line a strainer with a double layer of cheesecloth; set over a small bowl. Pulse ginger in a food processor until finely chopped. Transfer to prepared strainer. Gather edges of cloth together; squeeze tightly to release ginger juice into bowl. Scoop squash flesh into a food processor. Add sugar and 1 tablespoon ginger juice. With machine running, gradually add 1 1/2 cups cream and purée until smooth, adding more cream by tablespoonfuls if too thick. Season with salt, pepper, and more ginger juice, if desired. Gently warm in a saucepan, stirring constantly, until heated through. DO AHEAD: Can be made 1 day ahead. Gently rewarm purée in a saucepan, adding a little more cream if necessary.