Preheat the oven to 350 degrees F. Heat the olive oil in a large pot over medium-high heat. Add the carrots, onion, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the vegetables soften, about 5 minutes. Add the farro and cook, stirring, until toasted, 1 minute. Add the chicken broth and 3 cups water; cover and bring to a boil. Add the lemon zest, reduce the heat to medium low and simmer, uncovered, until the farro is almost tender, about 15 minutes. Stir in the chicken.
Whisk the eggs and lemon juice in a bowl; drizzle 1/2 cup of the hot broth into the eggs, whisking constantly. Gradually stir the hot egg mixture into the pot. Cook over medium-low heat, stirring constantly, until thickened, about 3 minutes. Season with 1/4 teaspoon salt.
Combine the feta and dill in a small bowl; season with salt and pepper. Fill the pita halves with the feta mixture and lightly brush with olive oil. Transfer to a baking sheet and bake until the cheese melts, 2 to 3 minutes. Top each serving of soup with more dill; season with pepper. Serve with the stuffed pitas.