In a large pot, melt 2 tablespoons of butter and 2 tablespoons of extra virgin olive oil over medium heat. Once melted, add 1 diced medium onion and 2 diced medium carrots. Sauté for about 3-4 minutes, stirring frequently until the vegetables soften and the onion becomes translucent.

Next, stir in 3 minced garlic cloves and 1 tablespoon of fresh rosemary. Cook these aromatics for an additional 30 seconds.

Pour in 6 cups of low sodium chicken broth, followed by 1 teaspoon of kosher salt. Bring the mixture to a boil.

Add 1 cup of orzo pasta into the boiling broth. Stir gently, then reduce the heat to maintain a steady simmer. Cook for 8-10 minutes, or until the orzo is tender.

While the orzo is cooking, whisk together 2 large eggs, 2 egg yolks, and the juice of 1 fresh lemon in a medium bowl.

To prevent curdling, slowly add a ladleful of the hot broth from the pot into the egg and lemon mixture, whisking continuously.

Return the soup to medium heat, then slowly stir in the tempered egg mixture while whisking gently. Cook for about 5 minutes.

Incorporate 2 cups of shredded rotisserie chicken, along with 1 tablespoon of fresh dill and the zest of 1 lemon. Stir everything together.

Ladle the soup into bowls, garnishing each serving with lemon slices, a sprinkle of fresh dill, and freshly ground black pepper.