Preheat oven to 400ºF, with one rack just below the middle position and the other just above it.
Peel the eggplants and cut them into small chunks—the pieces should be about the same volume to ensure they will all roast up in the same amount of time. Keep in mind that they will shrink somewhat during roasting. Put the pieces in a medium bowl.
Prepare the baking sheets. Add the olive oil to the bowl of eggplant and immediately stir with your hands to distribute the oil uniformly. Divide the eggplant between the two baking sheets. Add the shallots and garlic—just scatter them around—then immediately put the pans in the oven.
The eggplant will take about 40 minutes total to roast. At about 8 or 10 minutes, remove the baking sheets from the oven to toss everything and ensure things look good. At about 20 minutes, reverse the position of the pans in the oven, top to bottom. At about 30 minutes, stir everything again. If you think the eggplant is looking dry, add a bit more olive oil and quickly stir with a spatula.
While the eggplant is roasting, put the yogurt in a small bowl and stir in the fresh dill. Add salt and pepper to taste.
The eggplant is done when it is nicely browned and soft—around 35 to 40 minutes. Transfer the eggplant, along with the shallots and garlic, into a serving bowl. Add the yogurt sauce, gently stir everything together, check the seasonings, and that’s it. This is ready to serve.