Place the barberries in a fine-mesh sieve, rinse under cold water, then soak in warm water for 5 minutes; drain and pat dry.
Finely chop the parsley, cilantro, and dill. Thinly slice the scallions. Coarsely chop the walnuts.
Warm 1 tsp of the vegetable oil in a small skillet over medium-low heat. Add the barberries and cook, stirring, until glossy and slightly plumped, 30–60 seconds. Transfer to a plate to cool.
In a large bowl, whisk the large eggs with the kosher salt, ground black pepper, and ground turmeric until evenly colored and slightly frothy, about 30 seconds.
Fold the parsley, cilantro, dill, scallions, walnuts, and the cooled barberries into the eggs until evenly distributed.
Heat the remaining vegetable oil in a 10-inch nonstick or well-seasoned cast-iron skillet over medium heat until shimmering. Pour in the herb-egg mixture and spread it evenly, smoothing the top. Reduce heat to medium-low, cover, and cook until the edges are set and the bottom is deep golden, 8–10 minutes.
Run a spatula around the edges to loosen. Invert the kuku onto a plate, then slide it back into the skillet to cook the second side. Cook uncovered until the center is set and springy to the touch, 5–7 minutes more. As an alternative finish, place the skillet under a broiler to set the top, 2–3 minutes, watching closely.
Rest 5 minutes. Cut into wedges, garnish with the barberries (for garnish) and walnuts (for garnish), and serve with plain yogurt and flatbread. Serve warm or at room temperature.