Heat oven to 375°. Place eggplants on a foil-lined baking sheet, and prick all over with a fork. Bake, rotating, until skin is charred and flesh is very soft, about 1 hour; let cool. Scoop out flesh, transfer to a colander, and drain for 20 minutes.
Meanwhile, heat 1⁄4 cup oil in a 12″ skillet over medium-high heat. Add the medium onion; cook until browned, about 15 minutes. Add eggplant and minced garlic; cook, mashing, until liquid evaporates, about 5 minutes. Remove from heat; transfer to a bowl. Add yogurt, salt, and pepper; stir until smooth. Cool.
Meanwhile, heat 2 tbsp. oil in a 10″ skillet over medium-high heat. Add remaining onion; cook until caramelized, about 15 minutes. Transfer to a bowl; set aside. Wipe skillet clean and return to heat with remaining oil. Add sliced garlic; cook until lightly browned and crisp, about 3 minutes. Using a slotted spoon, transfer garlic chips to paper towels to drain; set aside. Transfer eggplant mixture to a bowl and drizzle with saffron; garnish with caramelized onions, garlic chips, and walnuts.