Preheat the oven to 425 degrees F.
Generously coat a heavy rimmed baking sheet with cooking spray. Place the beets on the baking sheet and spray them generously. Roast in the center of the oven until wrinkly and tender (a fork should slide through with no resistance), 45 to 60 minutes. Remove from the oven and set aside.
When the beets are cool enough to handle, peel them (wrap one at a time in a kitchen towel or wear gloves and gently massage the skin away). Slice the beets in half lengthwise. Place them flat-side down and cut each half lengthwise into 3 wedges. Keep the beets in separate bowls so the colors don't combine.
In a medium bowl, whisk together the vinegar, mustard, sugar, salt and pepper. While whisking, stream in the olive oil. Add about a third of the dressing to each of the bowls of beets and toss to coat.
In a separate bowl, toss the mint, arugula and remaining dressing to combine. Season with salt to taste.
To serve, place the salad mixture on a serving platter or on individual plates, top with the dressed beets and feta cheese and garnish with mint.