Heat olive oil in a large Dutch oven over medium-high heat until shimmering. Season chicken pieces generously with salt and pepper, then add to the pot in a single layer. Cook undisturbed for 4-5 minutes until golden brown on one side, then stir and cook 2 minutes more. The chicken does not need to be fully cooked through. Transfer to a plate with a slotted spoon, leaving the fond behind.

Reduce heat to medium. Add diced onion to the pot and cook for 5-6 minutes, scraping up the browned bits, until softened and translucent. Add minced garlic and cook 30 seconds until fragrant but not browned.

Return chicken and any accumulated juices to the pot. Add chicken broth, orzo, and bay leaf. Bring to a boil, then reduce heat to maintain a gentle simmer. Cook for 12-14 minutes, stirring occasionally, until orzo is tender and chicken is cooked through. Remove from heat and discard bay leaf.

In a medium bowl, whisk eggs until frothy. Whisk in lemon juice and zest until combined. Ladle 1 cup of hot broth from the pot into the egg mixture in a slow, steady stream while whisking constantly - this tempers the eggs so they do not scramble. Repeat with a second cup of hot broth.

Remove soup from heat. Slowly pour the tempered egg mixture back into the pot while stirring gently. The soup will turn creamy and pale yellow. Do not return to a boil or the eggs may curdle. Stir in chopped dill and nutmeg. Taste and adjust seasoning with salt and pepper.

Ladle into warm bowls immediately. Garnish with additional fresh dill and a thin lemon slice if desired. The soup will thicken as it stands; thin with a splash of hot broth or water if reheating.