Start by whisking together the yogurt, gherkins (and brine), Aleppo pepper (or gochugaru), sea salt, and chopped herbs in a medium bowl. Set aside while you prepare the rest of the salad.
Preheat your oven to 425°F (220°C). Meanwhile, bring a large pot of salted water to a boil. Once boiling, add the baby potatoes and simmer for 7-10 minutes, until just fork-tender. Drain well and let cool slightly.
Spread half the oil onto a large baking sheet, then arrange the potatoes in an even layer. Using the back of a jar or a glass, gently press down to flatten each potato slightly. Drizzle with the remaining oil, sprinkle with salt, and roast on the center rack for 27-30 minutes, flipping halfway through, until crispy and golden.
While the potatoes roast, combine the cucumber, radishes, and red onion in a large mixing bowl. When the potatoes have cooled slightly, use tongs to gently toss them with the salad and dressing, reserving a quarter of the potatoes to layer on top for extra crunch.
Finish with pistachios (or walnuts) and sliced red chili, layering the reserved crispy potatoes over the top. Serve immediately for the best texture.