Preheat oven to 425 degrees F. Scrub carrots and peel, if desired. Halve any large carrots lengthwise.
Line a 15x10x1-inch baking pan with foil. Toss carrots with olive oil. Evenly spread carrots in prepared pan. Sprinkle with teaspoon salt. Roast carrots for 15 minutes. Meanwhile, in a small microwave-safe bowl warm honey in microwave for 30 seconds. Whisk in curry powder; set aside.
Remove carrots from oven. Drizzle with honey mixture; toss to coat. Roast 10 minutes longer, turning occasionally, until carrots are glazed and tender when pierced with a fork. Transfer to a serving platter.
For Chive Yogurt, in a bowl combine yogurt, chives, and remaining teaspoon salt. Serve with roasted carrots.