Preheat oven to 400°F. Toss sourdough pieces with 3 tablespoons olive oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper on a rimmed baking sheet. Bake in preheated oven until golden brown, 8 to 10 minutes, stirring occasionally. Transfer croutons to a small bowl; set aside. Wipe baking sheet clean. (Do not turn oven off.)

Cut off and discard top 1/2 inch of garlic head. Place garlic head, cut side down, and onion wedges on rimmed baking sheet. Drizzle evenly with remaining 3 tablespoons olive oil, and sprinkle with 3/4 teaspoon salt and remaining 1/4 teaspoon pepper. Roast at 400°F until onions and garlic are tender and just starting to brown, 30 to 35 minutes. Pour wine over onions. Continue roasting until liquid is mostly evaporated, about 5 minutes.

Squeeze softened garlic cloves into a blender using tongs; discard garlic skins. Transfer onions and remaining liquid on pan to blender. Add chicken stock, Parmesan, and remaining 1/2 teaspoon salt. Secure lid on blender, and remove center piece to allow steam to escape. Place a clean towel over lid opening. Process on medium-high speed until smooth, about 2 minutes. Decrease speed to medium-low. With blender running, slowly pour in cream. Add butter, a few pieces at a time, and blend until soup is very smooth and creamy, about 30 seconds. Thin soup to desired consistency with additional chicken broth, if desired. Season to taste with additional salt.

Divide soup among serving bowls. Top with croutons; drizzle with additional olive oil, and garnish with additional black pepper.