Heat oven to 400°. In a large bowl, toss cauliflower and broccoli with 3 tbsp of the olive oil, 1/2 tsp of the salt and the black pepper. Distribute over 2 rimmed baking sheets in one layer. Roast at 400° for 15 minutes. Stir and roast 10 minutes more. If roasting at the same time as the Brussels sprouts, swap racks.
Meanwhile, heat remaining 1 tbsp olive oil in a sauté pan over medium heat. Stir in garlic and both paprikas; cook for 2 minutes. Scrape into yogurt and season with remaining 1/8 tsp salt. Drizzle veggies with spicy yogurt sauce.