Cook beets in a saucepan of boiling salted water until tender, about 1 hour. Drain, allow to cool slightly and slip off skins.
Place on grill and smoke for 60 minutes. Remove from grill, slice into 1/2-inch pieces and place in a heat proof bowl.
Bring remaining ingredients to a boil in a 2 to 3-quart saucepan, stirring until sugar is dissolved. Remove from heat and pour over cut beets. Cover with a plate to keep beets submerged in pickling liquid.
For the vinaigrette: Whisk together the 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 to taste.
Cook carrots in a large pot of boiling salted water for 1 minute. Strain and transfer to a bowl.
Set the temperature to 450 degrees F (set to 500 degrees F if using a WiFIRE-enabled grill) and preheat, lid closed 10-15 minutes.
Season carrots with salt and pepper. Toss carrots with 1-1/2 teaspoons oil and arrange on a baking sheet in a single layer. Place beets in same bowl and toss with remaining 1-1/2 teaspoons oil.
Add beets to baking sheet on opposite side of carrots. Place in preheated grill and roast, turning once, until just tender, about 15 to 20 minutes.
Place carrots, beets, 1/4 cup chopped parsley and feta cheese in a mixing bowl. Drizzle with vinaigrette and toss to coat. Arrange on a small platter. Sprinkle remaining chopped parsley over top. Enjoy!
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