Using a mandoline on the second or third setting, thinly slice the beets.
To segment the grapefruits, cut off the top and bottom ends of the peel, and working around the grapefruit remove the remaining peel and white pith by cutting down along the sides. To release the segments, cut between the membrane and segment and use the edge of the knife to release the segment from the other side of the membrane.
Layer the beets and grapefruit segments on a platter. Drizzle with olive oil, and aged balsamic vinegar, and sprinkle with salt and pepper.
Top with crumbled goat cheese and/or chives, if using. Serve immediately.