Heat a large saucepan over medium heat. Pour in the olive oil, and add the onion and garlic. Sauté for 3 to 5 minutes, or until the onion begins to turn translucent.
Add the turnips and potatoes and sauté for 2 minutes longer. Pour in the vegetable stock, bring to a boil then reduce the heat and simmer for 20 to 25 minutes, or until the turnips and potatoes are tender.
Transfer the soup contents to a blender and puree until completely smooth. Refrigerate to chill the soup completely and adjust the consistency with more stock or water. Add salt and black pepper.
To make the pesto, bring a small pot of salted water to a boil. Blanch the greens in the water for 10 seconds then transfer to an ice-water bath to cool. Place the greens into a clean towel and wring them to extract as much liquid as possible.
Coarsely chop the greens then add to a food processor or blender with the grated garlic and enough oil to get it moving. Add cheese and nuts, if using. Process to a paste, then season with lemon juice and salt and black pepper.
To make the crisp, whisk together the flour and egg whites until smooth. Drizzle the melted butter into the flour and egg white mixture while whisking. Allow the batter to sit for 30 minutes.
Preheat the oven to 350°. Spread the batter thinly and evenly onto a nonstick baking sheet. Sprinkle with poppy seeds and salt. Bake for 5 to 7 minutes, rotating halfway through baking. Remove from the oven and allow to cool.
To serve, ladle the soup into chilled bowls, drizzle pesto over the top and finish with a chunk of the poppy-seed crisp.