Make the soup: Preheat the oven to 375 degrees F. Cut the tops off the garlic heads to expose the cloves and remove any excess papery skins. Place on a large sheet of foil; drizzle with 1 tablespoon olive oil and sprinkle with 1/2 teaspoon each salt and pepper. Close the foil around the garlic, leaving a space between the tops and the foil, and place on a baking sheet. Roast until soft and fragrant, about 1 hour 15 minutes. Remove from the foil, let cool and squeeze out the garlic into a bowl.
In a large heavy soup pot (preferably enameled cast iron) over medium-high heat, melt the butter. Add the roasted garlic and cook, stirring, until slightly thick, about 5 minutes. Add the flour and cook until a dark golden roux forms, stirring frequently, 5 to 6 more minutes. Add 1 cup chicken stock and stir to incorporate, then repeat until all of the stock has been added. Add 2 cups water in 1-cup increments, stirring after each. Add the Italian seasoning, 1 1/2 teaspoons salt and 1 teaspoon pepper, and stir to combine well. Bring to a simmer and cook 15 minutes.
Meanwhile, make the crostini: Increase the oven temperature to 400 degrees F. In a small bowl, combine the olive oil, minced garlic, 1/4 teaspoon pepper, the paprika and oregano and mix well. Place the bread on a parchment-lined baking sheet and brush with the oil-spice mixture. Top each slice with about 1 teaspoon cheese and bake until the cheese is melted and the bread is starting to crisp, 8 to 10 minutes.
Add the cream to the soup and cook at a very low simmer, stirring occasionally, 10 more minutes. Adjust the seasoning if needed. Ladle into small mugs or bowls; top with parsley and drizzle with olive oil. Serve with the crostini.