Preheat the oven to 300 degrees F (150 degrees C).
Whisk olive oil, dill, and lemon juice together in a bowl. Add chicken and marinate in the refrigerator for 24 hours.
Chop marinated chicken into small cubes. Heat water and olive oil in a deep, covered saucepan over medium heat. Add chicken and cook until no longer pink inside, 5 to 7 minutes. Drain and allow to cool.
Heat 1/2 inch olive oil in a pan over medium-high heat. Saute eggplant until browned outside and soft inside, 3 to 5 minutes. Place on a baking sheet lined with paper towels to absorb extra oil. Allow eggplant to cool.
Chop cooled chicken into very small pieces and place in a bowl. Add ricotta cheese and Cheddar cheese. Whisk egg in a separate bowl and add to cheese mixture and mix well. Season filling with salt and pepper.
Place 1 1/2 tablespoons filling on the larger end of an eggplant slice. Roll around stuffing and place in the prepared baking dish. Add diced tomatoes on top of eggplant. Sprinkle with sugar. Drizzle with olive oil and season with salt and pepper. Cover with aluminum foil.
Bake in the preheated oven 40 minute to 1 hour. Remove from oven and cool 10 to 15 minutes before serving.