Preheat the oven to 400°F and line a pie pan with aluminum foil.
Peel and discard the papery outer layer of the heads of garlic. Keep the heads intact. (Do not detach the individual cloves.) Cut the tops of the garlic heads off, just enough to expose the tops of the cloves (about ½-inch). Place the garlic heads cut side up in the prepared pan. Drizzle with the oil, letting the oil soak in around the cloves. Top with another sheet of aluminum foil and seal the edges of the pan.
Bake 50-55 minutes, or until the garlic cloves are soft and deep golden brown. Set aside to cool.
Meanwhile, heat a large pot filled with salted water over medium-high heat. Add potatoes and boil until fork tender (about 8-10 minutes). Drain water.
Add potatoes and onions to the bowl of a blender. Squeeze all of the garlic cloves into the bowl and pulse until mixed. Add heavy cream, ¼ cup at a time, until the mixture is fully combined.
Add the garlic mixture back to the pot and set it over medium-low heat. (You don't want it to burn the cream.)
Whisk in chicken stock, parmesan, salt, pepper, and butter. Heat until everything is heated through, about 15-20 minutes, stirring occasionally.
To serve, cut a circle into the top of the bread bowl loaf and use your hands to scoop out the inside bread. Ladle your soup inside the bread bowl. Garnish with chives and croutons.