Preparation Stir cumin seeds in a small dry skillet over medium heat until fragrant, 3-4 minutes. Let cool. Finely grind in a spice mill. Whisk cumin, lemon juice, vinegar, and Dijon mustard in a medium bowl. Gradually whisk in 1/2 cup oil, then mint. Season vinaigrette with salt and pepper. Cook carrots in a large pot of boiling salted water for 1 minute; using a slotted spoon, transfer carrots to a bowl. Add beets to same pot of boiling water and cook until just tender, 20-25 minutes. Drain. Let stand at room temperature to cool slightly, about 15 minutes. Peel and trim beets; halve lengthwise. DO AHEAD: Vinaigrette and vegetables can be made 1 day ahead. Cover separately; chill. Return vinaigrette to room temperature and whisk before using. Preheat oven to 450°F. Season carrots with salt and pepper. Toss with 1 1/2 teaspoons oil. Arrange on half of a large rimmed baking sheet. Place beets in same bowl; season with salt and pepper and toss with remaining 1 1/2 teaspoons oil. Arrange in single layer opposite carrots on same baking sheet. Roast vegetables, turning once, until slightly charred and just tender, about 15 minutes. Place carrots, 1/2 cup parsley leaves, and feta in a medium bowl; drizzle 1/4 cup vinaigrette over and toss to coat. Arrange carrot mixture on a small platter. Place beets in another medium bowl. Add 2 tablespoons vinaigrette; toss to coat. Tuck beets in among carrots. Sprinkle remaining 1/4 cup parsley leaves over. Drizzle with remaining dressing.
Preparation Stir cumin seeds in a small dry skillet over medium heat until fragrant, 3-4 minutes. Let cool. Finely grind in a spice mill. Whisk cumin, lemon juice, vinegar, and Dijon mustard in a medium bowl. Gradually whisk in 1/2 cup oil, then mint. Season vinaigrette with salt and pepper. Cook carrots in a large pot of boiling salted water for 1 minute; using a slotted spoon, transfer carrots to a bowl. Add beets to same pot of boiling water and cook until just tender, 20-25 minutes. Drain. Let stand at room temperature to cool slightly, about 15 minutes. Peel and trim beets; halve lengthwise. DO AHEAD: Vinaigrette and vegetables can be made 1 day ahead. Cover separately; chill. Return vinaigrette to room temperature and whisk before using. Preheat oven to 450°F. Season carrots with salt and pepper. Toss with 1 1/2 teaspoons oil. Arrange on half of a large rimmed baking sheet. Place beets in same bowl; season with salt and pepper and toss with remaining 1 1/2 teaspoons oil. Arrange in single layer opposite carrots on same baking sheet. Roast vegetables, turning once, until slightly charred and just tender, about 15 minutes. Place carrots, 1/2 cup parsley leaves, and feta in a medium bowl; drizzle 1/4 cup vinaigrette over and toss to coat. Arrange carrot mixture on a small platter. Place beets in another medium bowl. Add 2 tablespoons vinaigrette; toss to coat. Tuck beets in among carrots. Sprinkle remaining 1/4 cup parsley leaves over. Drizzle with remaining dressing.