Step 1

Place a rack in lower third of oven; preheat to 425°. Blend garlic, harissa, 4 Tbsp. capers, and ¾ cup oil in a blender until a loose paste forms (a few chunkier pieces are okay); season with salt and pepper. (Alternatively, finely chop garlic and capers and mix well with harissa and oil in a small bowl.) Transfer ¼ cup paste to a small bowl and set aside.

Step 2

Score cut sides of eggplant flesh in a crosshatch pattern, being sure to stop before cutting through the skin below. Rub 1 Tbsp. paste remaining in blender evenly over each cut side of eggplant so entire surface is covered. Toss remaining 2 Tbsp. capers and 1 Tbsp. oil in a small bowl until capers are coated. Arrange eggplant, cut side up, on a rimmed baking sheet, leaving a little room on one side for capers; season with salt and pepper. Spread out capers in oil in a single layer in reserved space on baking sheet; set bowl aside.

Step 3

Roast eggplant and capers, keeping a close eye on capers, until capers are sizzling and slightly crisp, 8–10 minutes. Remove baking sheet from oven and transfer capers back to reserved bowl. Return eggplant to oven and continue to roast until tender, 10–12 minutes more for Japanese, 20–25 for globe. Let eggplant cool slightly, then slice in half crosswise on a diagonal.

Step 4

Mix cherry preserves into reserved paste in bowl. Mix yogurt and lemon juice in another small bowl to combine; season with salt and pepper.

Step 5

Arrange eggplant on a platter. Spoon lemon yogurt over, then drizzle harissa-cherry sauce on top. Scatter roasted capers, dill, and lemon zest evenly over eggplant.