Preheat the oven to 350 degrees F.
Remove the leaves and thick core from the cauliflower, coarsely chop, and reserve. Heat the olive oil in a large saucepan or soup pot over medium heat and add the onion and garlic. Cook until softened, but not browned, about 5 minutes. Add the cauliflower and stock and bring to a boil. Reduce the heat to a simmer, cover, and cook until the cauliflower is very soft and falling apart, about 15 minutes. Remove from heat and, using a hand held immersion blender, puree the soup, or puree in small batches in a blender* and return it to the pot. Add the Parmesan and stir until smooth. Season, to taste, with salt and black pepper. Keep warm until ready to serve.
Line a baking sheet with aluminum foil. Spread the shredded cheese over the foil in 1 even thin layer. Bake about 10 minutes until golden brown and crisps. Remove from oven and let cool 5 minutes. Break sheet of crisp cheese into large pieces and garnish each soup bowl with a couple shards and a pinch of fresh chives.
*When blending hot liquids: Remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.
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