Step 1Heat the oven to 375 degrees. Cut the 2 smallest zucchini in rounds about three-eighths-inch thick and put them side by side in a small roasting pan or a foil pan. Cut the remaining zucchini in one-fourth by one-fourth by one-half-inch dice. Put them in another small roasting pan in a shallow layer.
Step 2Drizzle the zucchini dice with 1 tablespoon olive oil and sprinkle with salt and pepper. Drizzle the rounds with 2 teaspoons olive oil, without seasoning them.
Step 3Roast the zucchini rounds and the dice for 10 minutes. Turn over each zucchini round; stir the diced zucchini. Continue to roast the zucchini until the rounds are just tender, about 7 more minutes, and the dice are very tender, about 15 more minutes.
Step 4Heat the broiler. Sprinkle the zucchini rounds lightly with paprika. Broil them, checking often, until they are lightly browned, about 3 minutes. Transfer the rounds to a plate and sprinkle with salt and pepper.
Step 5Transfer the zucchini dice to a medium bowl and add one-half teaspoon dried mint. Cool to room temperature.
Step 6Add the garlic, yogurt, labneh and fresh mint to the diced zucchini. Mix gently. Season to taste with salt and pepper. Refrigerate for about 1 hour to marry the flavors.
Step 7To serve, spread the zucchini-yogurt mixture on a plate or platter. Drizzle with olive oil if you like. Garnish by sprinkling a ring of dried mint about halfway to the center of the plate. If you like, sprinkle with Aleppo pepper or cayenne. Set a mint sprig in the center. Arrange zucchini rounds around the edge.