Preheat the oven to 400°F.
Remove two cloves from one of your bulbs and set them aside for later. Cut the tops off of all of your garlic bulbs. Place each bulb on a sheet of foil and drizzle with just enough oil to coat the outside, about ½ tablespoon. Loosely wrap each bulb in the foil.
Place the foil-wrapped garlic bulbs into the oven, either directly on the rack or on a baking sheet if you prefer, and roast them for about 40 to 45 minutes, until the garlic cloves are lightly browned and softened. Open the foil and let the garlic cool for a few minutes.
Once the garlic is cool enough to handle, squeeze the bulbs to extract the roasted garlic onto a small plate or bowl. Mince the two fresh garlic cloves that you reserved in step 1.
Coat the bottom of a large pot with a tablespoon of olive oil and set it over medium heat. Give the oil a minute to heat up, then add the shallots. Cook the shallots for about 5 minutes, stirring occasionally, until they soften.
Mince the two reserved fresh garlic cloves while your shallots cook.
Add the roasted garlic and the two minced fresh garlic cloves. Sauté the garlic briefly, stirring and mashing the roasted garlic with a fork, until the fresh garlic becomes very fragrant, about 1 minute.
Add the wine and stir to combine. Bring it to a simmer and let it cook until most of the liquid has evaporated and the mixture has thickened slightly, about 3 minutes.
Stir in the broth, bread, and thyme. Raise the heat and bring the liquid to a boil. Lower the heat to a simmer, then let the soup cook for about 20 minutes, until the bread has softened and is falling apart.
Remove the pot from heat. Blend the soup until smooth, either using an immersion blender, or by transferring it to a conventional blender. Always use caution when blending hot liquids.
Return the soup to the pot if you used a regular blender. Stir in the coconut milk and warm the soup back up over medium heat.
Ladle into bowls and optionally top each with a sprinkle of fresh parsley, croutons, and a drizzle of olive oil. Serve and enjoy!