In a Dutch oven or similar large, heavy-bottomed pot, melt the butter over medium-low. Add the onion and cook, stirring occasionally, until tender and turning translucent, 6 to 8 minutes. Adjust the heat as needed so the onion softens but does not brown. Add the garlic and cook until just until fragrant, about 30 seconds.

Add the broth, cauliflower, salt and pepper, and thyme. Increase the heat to high and bring to a boil. Lower the heat, cover, and let simmer until the cauliflower is tender when pierced with a fork, 10 to 15 minutes.

With an immersion blender, or working in batches with a high-speed blender (be careful! hot soup splatters; leave the center portion of the lid open and cover with a towel), puree the soup until smooth. Stir in the lemon juice and mustard, then the Parmesan. Carefully taste and adjust the seasoning as desired.

The soup will thicken as it cools, but if you’d like it thicker right away, you can partially cover the pot and simmer it for 5 to 10 minutes more. Ladle into bowls. Serve with a drizzle of oil and chopped fresh thyme or chives.