Move rack to top of oven and set broiler to high. Shave just the roots from onions, leaving core intact so that onion layers stay in place. Halve or quarter any larger onions. Arrange on a broiler pan and slide into oven. Broil until edges begin to char and onions are tender but still have some bite, about 10 minutes.
Meanwhile, set a Dutch oven or small stock pot over medium heat. Add olive oil, and when hot, add potatoes, 1/2 teaspoon sea salt, and a few twists pepper. Sauté potatoes until tender with crisp edges, about 7 - 10 minutes. Add white wine, stir 1 minute, and then add broth. Bring mixture to a boil, then turn heat to low. Taste and add sea salt as needed.
Pour back into soup pot and warm over medium-low heat just until mixture starts to steam. Heating soup gently is the best way to maintain its emerald green hue.
Ladle into bowls and garnish with shaved Parmesan, minced onion grass or chives, thinly slice green onions, and several twists white pepper.