Preheat oven to 450°F. Wrap beet tightly in aluminum foil; place on a rimmed baking sheet. Cook until tender when pierced with the tip of a sharp knife, 45 to 50 minutes. When cool enough to handle, peel (see below) and cut into wedges.
Meanwhile, prepare oranges: Slice off both ends of each with a paring knife. Cut away the peel and white pith, following the curve of the fruit. Holding the fruit over a bowl, cut along membranes to release whole segments. Squeeze juice from the membranes into another bowl, and add any accumulated juice from the segments.
Add oil, vinegar, and mustard to orange juice; season with salt and pepper, and whisk to combine. Add arugula, and toss to coat with dressing. Divide the arugula among four plates. Top with beet wedges, orange segments, and goat cheese. Serve immediately.