Finely chop the white part of the leek and transfer it to a colander. Thoroughly rinse the leek and set it aside.
Heat the olive oil in a large pot until shimmering. Add the leeks and a generous pinch of salt. Sauté the leeks over medium-low heat until very soft. About 10 minutes. Stir in the garlic and sauté for 30 seconds more.
Add the parsnip, beets, and vegetable stock. Increase the heat and bring the mixture to a boil. Cook until the parsnip is very tender. About 15-20 minutes. Taste and season with salt accordingly.
Transfer the vegetables to a food processor or blender. Blitz on low and slowly stream in the half and half. Once the mixture is smooth, transfer it to a bowl. Cover and let chill for 8 hours.
Preheat the oven to 400°F. Line a baking sheet with a silicone mat. Place mounds of shredded Parmigiano Reggiano on the mat, using a tablespoon as a guide. Sprinkle each cheese pile with freshly ground pepper and pop them in the oven. Bake for 3-5 minutes. Let cool slightly.
Carefully take the parmesan crisps off of the mat using a small offset spatula. Place on a cooling rack and let cool completely.
Take the soup out of the fridge 15 minutes prior to serving. Using a turkey baster, pour the soup into shot glasses. Top each serving with a small dollop of yogurt, micro greens, violets, and a parmesan crisp. Arrange the glasses on a platter and serve.