1. Melt 2 tablespoons of the butter in a large, heavy saucepan. Add the white mushrooms and cook over high heat, stirring occasionally, until golden, : about 8 minutes. Transfer to a plate.
2. Melt the remaining 4 tablespoons of butter in the saucepan. Add the onion and shallots and cook over moderate heat until softened. Add the wild mushrooms and cook, stirring occasionally, until tender, : about 20 minutes. Add the Cognac and cook until evaporated. Stir in the stock, thyme sprigs, bay leaves and cooked white mushrooms and bring to a simmer. Reduce the heat to moderately low and cook for 45 minutes. Discard the thyme sprigs and bay leaves.
3. Working in batches, puree the soup in a blender until smooth. Return the soup to the saucepan, stir in the cream and season with salt and pepper.
4. In a mini food processor, combine the Asiago, cream cheese and egg yolk and pulse until creamy. Spread the Asiago cream on the baguette toasts and arrange on a rimmed baking sheet.
5. Preheat the broiler. Broil the toasts : 8 inches from the heat for about 2 minutes, or until golden and bubbling.
6. Ladle the soup into shallow bowls and float the Asiago toasts on top. Garnish with the chives and serve.
Make Ahead: The soup can be prepared through Step 3 and refrigerated for up to 2 days. The toasts and Asiago cream can also be made ahead, but spread the cream on the toasts and broil them shortly before serving.