Heat oven to 425°F. Toss eggplant with 2 tablespoons oil, 1/4 teaspoon salt and pepper on a sheet pan and roast for 25 to 30 minutes, turning once, until browned and soft.
Bring rice, water and 1/4 teaspoon salt to a boil in a saucepan. Cover, reduce heat to low and cook for 15 minutes; rest off heat, covered, for 10 minutes.
Mix yogurt, squeezed cucumber, lemon juice, dill and the remaining 1/4 teaspoon salt. Refrigerate until serving.
Heat the remaining tablespoon oil in a large skillet over medium-high heat. Cook beef and onion for 8 to 10 minutes, breaking the beef into small crumbles, until browned and 160°F.
Add garlic, oregano and cinnamon and stir for 1 minute. Fold half the roasted eggplant into the beef, leaving the rest for the bowls.
Divide rice among four bowls and add beef, the remaining eggplant and cherry tomatoes. Spoon cucumber yogurt over part of each portion and serve.