Preheat oven to 350 degrees Fahrenheit.

Combine unpeeled garlic heads and unpeeled shallots in a large bowl.

Pour olive oil over the garlic and shallots, tossing to coat evenly.

Season generously with kosher salt and freshly ground black pepper.

Place garlic and shallots in a single layer in a baking dish, along with any remaining oil.

Cover the baking dish tightly with aluminum foil.

Bake for 35 minutes.

Check shallots for softness; remove if softened to prevent burning. If not soft, bake a little longer.

Continue baking garlic for another 30 minutes or so, until very soft.

Allow roasted garlic and shallots to cool slightly.

While garlic and shallots are cooling, melt butter in a heavy 3- to 4-quart saucepan.

Add coarsely chopped yellow onions to the saucepan and cook gently for 15 minutes, being careful not to brown them.

Add peeled, unroasted shallots and peeled garlic cloves to the saucepan.

Cook over low heat for 10 minutes longer.

Add quartered russet potatoes, salt (about 1/2 teaspoon or to taste), chicken stock, heavy cream, and fresh thyme leaves to the saucepan.

Cover the saucepan and cook for approximately 40 minutes, or until potatoes are very soft.

Peel the roasted shallots.

Cut roasted garlic heads in half horizontally and squeeze the soft pulp from the garlic cloves into a potato ricer or by hand to extract the pulp.

Add the garlic pulp and peeled roasted shallots to the saucepan with the soup.

Cook for 10 minutes longer to allow the flavors to meld.

Carefully puree the soup in a blender or food processor until smooth.

Season the pureed soup to taste with salt and pepper.

If desired, adjust the consistency of the soup by adding additional stock, cream, milk, or water until the desired thickness is reached.

Reheat the soup before serving.