Preheat your oven to 400°F and line a large baking sheet with parchment paper. Toss cauliflower florets with olive oil, sprinkle generously with salt and pepper, then spread in an even layer on the prepared pan. Roast for about 25 minutes until the edges are golden brown and tender when pierced with a fork.
While the cauliflower roasts, melt butter in a large Dutch oven over medium heat. Add the diced apple and onion, cooking until softened and beginning to caramelize, about 8-10 minutes. Stir in garlic and thyme, cooking for another minute until fragrant.
Pour in vegetable broth and bring to a simmer. Add the roasted cauliflower (along with any juices from the pan). Cook for 5 more minutes so flavors meld together.
Using an immersion blender (or carefully transferring to a standard blender in batches), blend until completely smooth — about 2-3 minutes. Stir in cream or milk and season with salt and pepper to taste. The soup should coat the back of a spoon; if it's too thick, add more broth a splash at a time.
For serving, heat a small pan over medium-high heat and melt 1 tablespoon butter until it starts foaming and turns golden brown (about 2 minutes). Ladle soup into bowls and drizzle each with the nutty browned butter.
