Melt the butter in a large pot over medium heat. Add the onion and cook, stirring occasionally, until softened, about 5 minutes. Stir in the garlic and cook 1 minute more.

Add the potatoes, broth, salt, pepper, and thyme. Bring to a boil, then reduce heat to medium-low and simmer, partially covered, until the potatoes are fork-tender, 15 to 20 minutes.

Remove the pot from the heat. Use an immersion blender to blend just until about half the potatoes are puréed, leaving plenty of chunks. (Or transfer 2 cups of soup to a blender, purée, and return to the pot.)

Return the pot to low heat. Stir in the heavy cream and warm through, about 2 minutes. Taste and add more salt and pepper if needed.

Ladle the soup into bowls. Drizzle with a little extra cream and top with chopped chives.