Add the cauliflower, garlic and 1/4 cup of olive oil to a rimmed baking sheet. Sprinkle generously with salt and pepper and toss to coat. Roast until the cauliflower is fork tender and lightly charred in places, about 25 minutes.
In the meantime, heat a Dutch oven over medium-high heat. Add the remaining 2 tablespoons of olive oil, potatoes, celery, onion and thyme. Sprinkle with a generous pinch of salt and pepper. Cook, stirring often, until the vegetables have slightly softened, about 5 minutes. Add the white wine to deglaze the pan, scraping up any brown bits that have developed at the bottom of the pan. Add the chicken broth, Parmesan rind and roasted cauliflower and garlic. Bring to a boil, then turn the heat down to low and cover. Simmer for 25 minutes, stirring occasionally to prevent the cheese rind from sticking to the bottom of the pan. Remove the cheese rind and discard.
In batches, carefully transfer the mixture to a high powered blender. Blend until completely smooth (see Cook's Note).
Once all the soup is blended, add it back into the Dutch oven. Season with salt and pepper to taste. Ladle into bowls and garnish with a pinch of pepper and a sprinkle of Parmesan.
Do not overfill the blender and be sure to vent slightly.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.