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.
Meanwhile, make the Parmesan crisps:
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.