Fire up your grill for high-heat direct cooking. You want to get your grill to 450 F or higher. Alternatively, you can roast the beets in the oven set to 450 F.
Place the beets on a large piece of aluminum foil.
Add the ice and seal the foil tightly.
Place over direct heat and cook for 45 minutes or until the beets are fairly tender when poked with a knife or skewer.
Remove beets from grill and carefully open the foil.
Let cool slightly then remove the skin from the beets and slice.
Place beets in a medium bowl with the oil, 1 tablespoon of the vinegar, and salt and pepper to taste. Stir gently to combine.
In another bowl, gently combine the cucumber, remaining 2 teaspoons of vinegar, sour cream, dill, and salt and pepper to taste.
Serve by placing beets on a dish, then add the cucumbers on top.