1. Preheat oven to 350 degrees.

2. Place beets in a large baking dish, setting aside the prettiest beet; drizzle beets in baking dish with 2 tablespoons olive oil and 1 tablespoon vinegar; season with salt and pepper. Cover with parchment-lined aluminum foil and transfer to oven. Bake until tender, about 2 hours. Remove from oven and let cool.

3. Meanwhile, prepare an ice-water bath. Thinly slice reserved beet and carrots crosswise on a mandoline; transfer to ice-water bath and let stand 1 to 2 minutes. Drain and set aside.

4. Slice the top and bottom of whole grapefruit with a paring knife. Set one of the flat sides on your work surface. From top to bottom, following the curve of the fruit, cut away peel and white pith. Trim off any remaining pith. Holding the grapefruit over a bowl, cut along both sides of each segment, staying close to the membrane, to release. Discard membrane. Set grapefruit slices aside.

5. In a small bowl, whisk together grapefruit juice, garlic, and remaining 2 tablespoons vinegar. Whisk in remaining 1/4 cup olive oil until emulsified; set vinaigrette aside.

6. Once beets are cool enough to handle, use a paring knife to peel away skin. Place peeled beets on the rack of on indoor smoker or a wire rack if using an outdoor grill. If using an indoor smoker, preheat a grill pan over low heat in a well ventilated area. Place wood chips in smoker and top with rack. Carefully light wood chips; cover and place on grill pan. If using a grill, preheat to low with a small fire lit off to one corner. Place wire rack on grill and top charcoal with wood chips; close grill. Let beets smoke for 20 minutes.

7. Transfer smoked beets to a large bowl along with sliced beets and carrots, grapefruit slices, arugula, herbs, and pistachios. Drizzle with vinaigrette and toss to combine. Season with salt and pepper and garnish with cheese. Serve.