Heat the oil in a large heavy pot or Dutch oven over medium-low heat. Add the onions, the smashed garlic cloves, bay leaves, salt, and pepper. Cook, stirring often, until the onions are softened and golden, 10 to 15 minutes. Add the minced garlic and stir until fragrant, about 1 minute. Stir in the stock and thyme and bring to a boil. Reduce the heat to medium-low. Simmer, stirring occasionally, for 1 hour.

Add the bread cubes. Cook, whisking often, until the bread disintegrates and thickens the soup, about 10 minutes. Remove and discard the bay leaves. Purée the soup with an immersion blender or in batches in a food processor or blender. Return to the pot and whisk in the cream, cheese, and parsley. Season with additional salt and pepper if necessary.

Ladle into bowls. Serve with the hot cheese toasts.