Prepare the squash by peeling, removing the seeds, and cutting it into 1-inch chunks. Thinly slice the onion.
Melt the butter in a medium pot over medium heat. Add the sliced onion and cook, stirring, until translucent and soft but not browned, 6–8 minutes.
Add the kabocha squash to the pot and stir to coat with the butter and onions, 1–2 minutes.
Pour in the vegetable stock and bring to a boil over high heat. Reduce to a gentle simmer, cover partially, and cook until the squash is very tender and easily smashed with a spoon, 15–20 minutes.
Remove from the heat. Blend the soup until completely smooth using an immersion blender, or carefully in batches in a countertop blender; return the soup to the pot if needed.
Stir in the whole milk and the 0.5 cup heavy cream. Warm gently over low heat until steaming but not boiling, 3–5 minutes.
Season with 1 tsp of the kosher salt and the white pepper. Taste and stir in up to the remaining 0.25 tsp salt as needed.
Ladle into bowls and garnish with a drizzle of heavy cream and a sprinkle of snipped chives if desired. Serve hot.