01 - Preheat oven to 400°F. Cut the tops off both garlic heads to reveal the cloves, drizzle with 1 tablespoon olive oil, wrap tightly in foil, and roast for 25 minutes until the cloves are tender and golden.
02 - While garlic roasts, heat the remaining 1 tablespoon olive oil in a large pot over medium heat. Add diced onion, chopped celery, and chopped carrot; sauté for 5 to 6 minutes until vegetables are softened.
03 - Add peeled and diced potatoes, thyme, bay leaf, and vegetable broth to the pot. Bring to a boil, reduce heat, and simmer for 15 minutes until potatoes are tender.
04 - Squeeze roasted garlic cloves from their skins directly into the pot. Remove and discard the bay leaf.
05 - Puree the mixture using an immersion blender, or transfer in batches to a blender, until the soup reaches a smooth consistency.
06 - Stir in milk and season with salt and freshly ground black pepper. Gently reheat without allowing to boil.
07 - Ladle hot soup into bowls, garnishing with chopped parsley and offering crusty bread or croutons if desired.