Hollow out the zucchini using a zucchini corer or a small spoon, leaving about 1/4 inch of flesh all around. Set aside the scooped flesh for another use or discard.
In a mixing bowl, combine the uncooked rice, ground meat, onion, pine nuts, allspice, cinnamon, black pepper, and salt. Mix well until the ingredients are evenly distributed.
Fill each zucchini with the rice and meat mixture, leaving about 1/2 inch at the top to allow the rice to expand. Do not overstuff.
Heat the olive oil in a large pot over medium heat. Arrange the stuffed zucchini in the pot and cook for a few minutes until they start to brown slightly. Pour the broth over the zucchini, cover the pot, and simmer for about 1 hour, or until the rice and meat are fully cooked.
While the zucchini is cooking, combine the Greek yogurt, minced garlic, dried mint, and salt in a bowl. Stir until well blended and set aside.
Once the zucchini is cooked, carefully remove them from the pot and place them on a serving dish. Serve hot with the yogurt sauce on the side.