Prepare cauliflower: Wash and cut cauliflower into small pieces, including the stock. Cut thick parts smaller to ensure even cooking.
Make roux: In a pot, melt 3 tablespoons of unsalted butter over medium heat. Add 1/4 cup all-purpose flour and stir continuously to form a roux. Cook until it turns a light brown color, being careful not to burn it.
Add water and bring to boil: Slowly add 5 cups of water while stirring to avoid lumps. Use a whisk if needed to smooth out any chunks. Increase heat to high and bring the mixture to a boil.
Add flavorings and cauliflower: Add 1 tablespoon beef base, 1 teaspoon salt, chopped cauliflower, and freshly grated nutmeg to the pot. Stir well.
Simmer cauliflower: Cover the pot with a lid slightly ajar to allow steam to escape. Reduce heat to medium and maintain a gentle boil. Cook for 10 minutes, stirring occasionally.
Prepare egg: While soup cooks, whisk one egg thoroughly until smooth with no large pieces of egg white.
Blend soup: Scoop out some cauliflower pieces to set aside if you prefer texture in the soup. Use an immersion blender to blend the remaining soup until mostly smooth but still with some cauliflower pieces left.
Add cream and reheat: Add 1 cup of half and half to the blended soup and warm it up over low heat until it reaches a gentle boil.
Add egg and reserved cauliflower: Slowly pour the whisked egg into the soup while stirring continuously. Cook for about 30 seconds. Then add the reserved cauliflower pieces back into the soup and warm through.
Serve: Ladle soup into bowls, garnish with chopped parsley, and serve hot.