Heat 1 1/2 tablespoons of the oil in a large skillet over medium-high. Add half of the mushrooms, and cook, stirring occasionally, until lightly browned, about 5 minutes. Transfer mushrooms to a plate. Repeat procedure with 1 1/2 tablespoons of the oil and remaining mushrooms. Reduce heat to medium, and add onion, garlic, thyme, and remaining 2 tablespoons oil; cook, stirring often, until softened, about 5 minutes. Add wine; cook, stirring constantly, until wine is absorbed, about 90 seconds. Stir in cauliflower, water and stock. Cover and cook, stirring occasionally, until cauliflower is crisp-tender, 10 to 12 minutes. Remove from heat.
Transfer 3 cups of the cauliflower mixture along with any remaining liquid in the skillet into a blender. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over opening. Process until smooth, about 15 seconds.
Return skillet with cauliflower to medium heat. Stir in cauliflower puree, mushrooms, salt, pepper and 1/4 cup of the cheese. Cook, stirring constantly, until cheese is melted and mixture has a creamy consistency, about 1 minute. Divided risotto among 4 bowls; sprinkle evenly with remaining 1/2 cup cheese. Garnish with thyme leaves and serve immediately.