Preheat oven to 400 F. Line a baking sheet with aluminum foil.
On the prepared baking sheet, toss the eggplant with 1/4 cup of the olive oil and a pinch of salt to coast, then turn the halves cut-side down. Roast for 30 to 35 minutes, until tender. Let the eggplant halves cool slightly, then using a spoon scoop out the flesh into a food processor and discard the skins.
While the eggplant roasts, in a large skillet, heat the remaining 1/4 cup olive oil over medium heat. Add the onions and garlic and cook, stirring often, until softened and caramelized, 25-30 minutes. Stir in the tomato paste and cook, stirring continuously, until incorporated and caramelized to the colour of rust, about 5 minutes. Add the cumin, turmeric, and cayenne and cook, stirring until fragrant, about 2 more minutes.
Transfer the onion mixture to the food processor with the eggplant and pulse until well incorporated, keeping the mixture chunky. Taste and adjust seasoning with salt.
Transfer the dip to a bowl and top with yogurt, swirling with a spoon. Garnish with chopped walnuts and fresh mint. Serve with toasted pita.