Preheat your oven to 450°F and place a baking sheet on the center rack to heat. In a large bowl, toss the halved potatoes with 2 tablespoons of olive oil, ½ teaspoon salt, and ½ teaspoon pepper until evenly coated. Carefully spread the potatoes in a single layer on the hot baking sheet. Roast for about 35 minutes, or until golden and tender. Remove from oven and let cool slightly.

While the potatoes roast, grate one cucumber half using the large holes of a box grater. Wrap the grated cucumber in a clean kitchen towel and squeeze over the sink to remove excess moisture. Transfer to a medium bowl and mix in the yogurt, lemon juice, 1 tablespoon olive oil, dill, mint, grated garlic, and ¼ teaspoon salt until well combined.

Chop the remaining cucumber half and place it in the reserved potato bowl. Add chickpeas, cherry tomatoes, sliced red onion, 2 tablespoons olive oil, and the remaining ¼ teaspoon salt. Toss gently to combine.

To assemble, spoon the yogurt tzatziki evenly into 4 shallow bowls. Top each with a portion of roasted potatoes and the chickpea-vegetable mixture. Drizzle the remaining 1 tablespoon olive oil over the bowls and garnish with extra dill, if desired.