Drizzle beets with 1 tablespoon olive oil and wrap in tin foil. Bake for 25-35 minutes, depending on size, until a fork pierces through it easily. Let cool completely.
Meanwhile, cook farro according to packaging instructions. Let cool completely.
In a small bowl, whisk together 3 tablespoons olive oil, balsamic vinegar, and lemon juice. Set aside.
Once beets are cool, cut in half and slice with a mandolin into thin rounds. In a large bowl, toss beets, carrots, farro, scallions, chives, and feta. Pour in dressing and toss to coat. Season with salt and pepper to taste.