Cut off the broccoli florets and chop them. Trim off the bottom 2 inches of each stem, peel them, and thinly slice. Combine the broccoli, oil, garlic, thyme, broth, wine, and lemon juice in a slow cooker. Cover and cook on LOW for 5 to 6 hours.
Puree the soup with a handheld immersion blender or transfer to a food processor or blender and puree in batches. Season with salt and pepper. Serve in bowls, passing the toast and Parmesan cheese.