Heat a large stockpot or Dutch oven over medium heat. Add the oil and butter. Swirl the pan until the butter is melted. Add the onion and carrots and stir to combine. Cook for 3-4 minutes until the onion is softened. Add the garlic and rosemary and stir. Cook for another 30 seconds.
Return the pot to the heat now and add the egg mixture. Cook over medium-low heat for another 5 minutes, stirring constantly until soup is thickened a bit. Add the chicken, fresh dill and lemon zest. If you like a thinner soup, add more chicken broth. Taste and add more salt and/or lemon juice, if needed. Serve garnished with lemon slices, freshly ground black pepper and dill sprigs.
See Café Tips for more detailed instructions and tips.