Peel the garlic cloves and toss them with olive oil. Spread on a baking sheet and roast in a preheated 400°F (200°C) oven until fragrant and golden, about 15 minutes.
While the garlic roasts, peel and cube the potatoes into evenly sized pieces for quick and uniform cooking.
Heat a large pot over medium heat, then add the roasted garlic and a drizzle of olive oil. Sauté until the garlic is fragrant, about 1 minute.
Add the cubed potatoes to the pot and stir to coat them with the garlic and oil. Cook for another 2-3 minutes.
Pour in the vegetable broth and bring to a boil. Reduce heat to low, then cover and simmer until the potatoes are tender, about 15 minutes.
Use an immersion blender or transfer the soup to a blender to blend until smooth and creamy. Be careful blending hot liquids—blend in batches if necessary.
Return the blended soup to the pot if using a blender. Stir in the chopped thyme and heavy cream for extra richness, then season with salt and pepper to taste.
Heat the soup gently over low heat just until warmed through, then taste and adjust seasoning if needed.
Serve the soup hot, garnished with additional thyme or a drizzle of cream if desired. The soup should be velvety with tender potato chunks visible on the surface.