Preheat the oven to 400°F (200°C). Wash the eggplants thoroughly and pat them dry. Slice each eggplant in half lengthwise. Place them cut-side up and use a sharp knife to score the flesh, making deep lengthwise cuts followed by diagonal cuts to create a diamond pattern.

Line a baking sheet with parchment paper and lightly brush it with olive oil. Drizzle additional olive oil over the cut surfaces of the eggplants, ensuring they are well-coated, then sprinkle with salt. Arrange the eggplants cut-side down on the prepared baking sheet. Roast for 40 to 45 minutes, or until the bottoms turn golden and caramelized while the flesh becomes soft. A knife should glide easily through the skin when tested.

Once roasted, remove from the oven and transfer to a serving plate. In a small bowl, mix olive oil, minced garlic, and fresh basil to create the garlic-infused oil. Generously brush this fragrant mixture over the roasted eggplants.

For the yogurt dip, whisk together yogurt, salt, lemon juice, olive oil, and dill in a separate bowl until well combined. Serve alongside the warm eggplant and enjoy immediately.