Heat the olive oil in a large pot over medium heat. Add the garlic and cook for 5 minutes to soften. Add the potatoes 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. Taste for seasoning and season with additional salt and pepper if desired. 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 eggs. Add the egg whites to the pot, stirring for 1 minute. Add the egg yolks 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. Sprinkle the chopped squash blossom on top of the soup, and drizzle with a little bit of olive oil. Serve immediately.