In a large pot, heat 2 tablespoons of olive oil. Add the leek, fennel bulb and fronds, and a generous pinch of salt. Cook over moderate heat until tender, but not browned, five to seven minutes. Add the garlic and cook until tender, about one minute.
Add the arugula, dandelion greens (or spinach), romaine, vegetable broth, and cannellini beans, and bring to a boil over moderate heat. Partially cover, reduce the heat to low and simmer until greens are wilted, about five minutes. Season with salt and pepper. Remove the pot from the heat and stir in dill and lemon juice. Using an immersion blender (or working in batches in a blender and returning to the pot), puree the soup until smooth. Keep warm over low heat.
Meanwhile, fill a deep skillet with water and bring to a simmer. Add the vinegar, then, one at a time, break the eggs into a small bowl and gently ease them into the simmering water. Cook until the whites are just firm and the yolks are still runny, about three minutes. Using a slotted spoon, transfer the eggs to a paper towel-lined plate.
To serve, divide the soup into six bowls and top each with a poached egg. Drizzle with olive oil and sprinkle with flaky sea salt and more black pepper, as desired. Garnish with dill sprigs.