Heat the olive oil in a large pot over medium heat. Add the garlic and cook for 5 minutes to soften. Add the tomato paste, tomato, chili pepper, oregano, thyme, basil, rosemary, and sage and cook for 4 minutes longer. Add the water, season with salt and pepper, and bring to a boil. Reduce the heat to low and simmer for 30 minutes. Remove from the heat.
Transfer the soup to a blender and, carefully, puree the soup for 1 minute, or until it has a smooth consistency. Return the soup to the pot over medium-low heat.
Divide the egg. Add the egg white to the pot, stirring for 1 minute. Add the egg yolk to the pot and stir until combined. Add the vinegar and stir. Taste for seasoning and season with additional salt and pepper, if needed.
Divide the soup into individual bowls. Arrange some chili peppers on top of the soup, if using, and garnish with a thyme sprig. Serve immediately.