Preheat the oven to 400°F. Line a baking sheet with parchment paper.
Using a spoon, scoop out a third of the flesh inside the eggplants (you can set it aside for other uses or discard). Rub the inside of each eggplant half with 1½ teaspoons olive oil and transfer to the prepared baking sheet.
Add the remaining 1 tablespoon olive oil to a large skillet and heat over medium heat. Add the onion and sauté until tender, 3 to 4 minutes. Add the garlic and cook until fragrant, 1 minute more.
Add the mushrooms and cook until they are just tender, 4 to 5 minutes. Stir in the kale and quinoa, and cook until the kale is wilted slightly, 2 to 3 minutes. Season the mixture with thyme, lemon zest and juice, salt, and pepper.
Spoon the filling into the prepared eggplants and roast until the eggplants are tender but not falling apart, 17 to 20 minutes. Let cool for 5 minutes.
Serve the eggplant immediately, garnished with parsley or basil and accompanied by the yogurt and additional lemon wedges.