Preheat the oven to 350°F (180°C). Prepare hot water for a water bath.
Combine water and sugar in a small, heavy-bottomed saucepan. Cook over medium heat, without stirring, until the sugar turns medium to dark brown. Carefully pour the hot caramel into a 6-inch round cake pan, tilting to coat the bottom evenly. Set aside to cool and harden.
Cut the kabocha squash into 2-inch (5 cm) cubes. Steam the cubes for about 15-20 minutes, or microwave for 8-10 minutes, until tender when pierced with a fork. Allow the cooked squash to cool until safe to handle. Remove the skins by gently scraping the flesh with a spoon. Mash the flesh, then press it through a mesh strainer to create a smooth puree. Set the puree aside.
In a large bowl, whisk together eggs and sugar until well combined. In a saucepan, heat milk and cream over medium heat until it just starts to bubble. Slowly pour the hot milk mixture into the egg mixture in a thin stream, whisking constantly to temper the eggs. Continue whisking vigorously throughout this process to prevent curdling.
Add the kabocha puree and vanilla to the egg and milk mixture, whisking constantly to incorporate fully. Carefully pour the kabocha custard mixture into the caramel-lined cake pan.
Place the filled cake pan in a larger baking dish or roasting pan. Carefully pour hot water into the larger pan until it reaches about halfway up the sides of the cake pan, creating a water bath (bain-marie). Carefully transfer the water bath setup to the preheated oven. Bake for 35-40 minutes. To check for doneness, insert a knife into the center of the purin. If it comes out clean without liquid clinging to it, the purin is done.