Preheat the oven to 375ºF.
With a sharp knife, slice off the top layer of the garlic head/s to expose most of the cloves and peel off as many outer layers of the garlic as possible. You should still be left with a primary layer of skin around the cloves.
Drizzle the exposed cloves with olive oil and sprinkle with salt and coarse pepper. Wrap the garlic head/s in a piece of aluminum foil and place the foiled bundle/s on a baking sheet.
Roast in the oven for 40 minutes or so until the garlic is gorgeously fragrant and, once carefully unwrapped, has taken on a golden color and butter-soft texture.
To extract the delicious caramelized garlic from the skin, simply work from the bottom and gently squeeze the garlic through the skin (much like toothpaste) the garlic will ooze out. You must sample (but try not to eat it all!) set the warm roasted garlic aside.
In a large pot or Dutch oven, melt butter over low-medium heat. Add the onions and sauté for 5 minutes or so until translucent. Sprinkle the buttered onions with flour, mixing well to integrate and make sure that the flour doesn’t brown/burn while it’s mixing.
Add the broth (starting with 4 cups), milk, cream, thyme and a few cracks of salt & pepper, whisking to combine. Allow the soup to come to a boil, while continuing to whisk for 10 seconds or so, and then reduce to lowest heat and allow the broth to simmer for another couple minutes as the roux works its magic and the soup begins to thicken somewhat. Turn off heat.
Add the cooked potatoes and roasted garlic cloves intermingling with the béchamel sauce.
Working in batches, carefully transfer ingredients to a blender or food processor and blend until completely smooth. For a more dilute consistency, add a little more broth to desired consistency. Taste the soup and adjust seasonings as desired.
Meanwhile, in a separate pan, sauté mushrooms in a little bit of butter or olive oil until a nice sear develops, adding a few cracks of salt/pepper as desired. Be sure to allow the pan to heat up first and allow the natural water contained in the mushrooms to be released and eventually burnt off through the cooking process. Set the seared mushrooms aside to cool slightly before adding the majority of them to the blended soup (while retaining a few to garnish the top of the soup).
Serve the soup warm and topped as desired with fresh thyme sprigs, green onion/chives, black pepper, seared mushroom and/or cheese.