For the yogurt-herb sauce: Whisk yogurt, cilantro, mint, and garlic together in bowl until combined and season with salt and pepper to taste. Cover and refrigerate until ready to serve, at least 30 minutes.
For the cauliflower cakes: Adjust oven rack to middle position and heat oven to 450 degrees. Toss cauliflower florets with 1 tablespoon oil, turmeric, coriander, salt, ginger, and pepper. Transfer to aluminum foil–lined rimmed baking sheet and spread into single layer. Roast until florets are well browned and tender, about 25 minutes. Let cool slightly, then transfer to large bowl.
Line clean rimmed baking sheet with parchment paper. Mash florets coarsely with potato masher. Stir in goat cheese, scallions, egg, garlic, and lemon zest until well combined. Sprinkle flour over cauliflower mixture and stir to incorporate. Using wet hands, divide mixture into 4 equal portions, pack gently into 3/4‑inch-thick cakes about 31/2 inches in diameter, and place on prepared sheet. Refrigerate cakes until chilled and firm, about 30 minutes.
Line large plate with paper towels. Heat remaining 3 tablespoons oil in 12‑inch nonstick skillet over medium heat until shimmering. Gently lay cakes in skillet and cook until deep golden brown and crisp, 5 to 7 minutes per side. Drain cakes briefly on prepared plate. Serve with yogurt sauce and lemon wedges.