In a large soup pot, add 5 cups of water and bring it to a boil. Add the lobsters (head down) and cook them for 10 minutes. Use a strainer to remove the lobsters and let them cool.
Extract the lobster meat from the tails and claws. Cut the lobster meat into bite-sized pieces and set aside.
Add the lobster heads and shells back to the boiling water. Simmer over medium-low heat for 30 minutes, or until the stock reduces to 4 cups. Strain the lobster stock and discard the heads and shells.
In a medium soup pot, heat the butter and sauté the onion, carrot, and tomato over low heat until the vegetables begin to caramelize.
Add the lobster stock, white wine or Japanese cooking sake, brandy or sherry (if using), turmeric powder, bay leaves, and Italian parsley sprigs. Simmer the bisque over medium-low heat for 30 minutes.
Discard the bay leaves and Italian parsley sprigs from the soup. Using a blender or immersion blender, purée the soup until smooth. Strain the soup through a fine mesh sieve and discard the residue.
Simmer the soup again over low heat. Add the whipping cream and season with sea salt, freshly ground black pepper, cayenne pepper, and mirin (if using). Thicken the soup with a cornstarch and water mixture. Simmer on low heat for 15 minutes, or until the soup reaches your desired thickness.
Add the lobster meat to the soup and stir to combine well. Turn off the heat and ladle the lobster bisque into four soup bowls. Serve immediately, garnished with chopped parsley.