Preheat the oven to 350 degrees.
Heat the oil in a pot over medium heat, add the onion and carrot and cook for about 5 minutes or just until the carrot is tender and the onion is translucent.
Season the chicken with salt and pepper.
Add the rice and chicken broth to the pot and bring to a boil. Reduce the heat to a low simmer, add in the chicken and simmer until the rice is cooked through, about 15 - 20 minutes.
Meanwhile in a small bowl beat the egg and lemon juice together.
Stir in 1 -2 ladles of the broth from the soup to the bowl with the egg and lemon, this is to temper the egg/lemon mixture. Pour the contents of the bowl back into the simmering soup and stir. You will notice the soup thickening up. Adjust the seasoning with salt and pepper to taste and if you feel the soup needs more lemon, add more lemon juice as desired.
Drizzle the olive oil on the pita bread and season with salt and pepper, place on a baking sheet and warm in the oven for 3 - 4 minutes.
Remove the soup from the heat and stir in the dill. Garnish with lemon slices. Serve warm pita on the side.