Heat the vegetable stock in a small pan and set it aside.
In a medium sized pan, gently heat the oil, then add the onion, and allow to sweat on medium heat, until the it is soft.
Add the garlic, lovage stems and half of the chopped lettuce (the part form the core). Lower the heat and allow to cook for further 5 minutes.
Add the stock and yogurt and gently whisk to combine them together. Season with salt and bring to a gentle boil. Finally add the remaining chopped lettuce leaves and continue to boil for about 2-3 minutes. In the end, add the chopped lovage leaves and dill.
Separately, poach the eggs and set them aside.
To serve, place a generous amount of broth in a bowl, add the poached egg and season with lemon juice and freshly ground black pepper