Preheat the oven to 400 degrees F.
Wrap the beets in aluminum foil and roast in the oven until they are easily pierced with a paring knife, about 30 to 45 minutes. Remove from the oven and let cool. When the beets are cool enough to handle, peel them and cut into 1/2 moon shapes about 1/4-inch thick. Set the beets aside to cool completely.
Preheat the grill or broiler.
In a medium-sized mixing bowl, combine the beets, vinegar, oil, caraway seeds, chives, and salt and pepper.
Grill or toast the bread until crispy and golden brown. Top each slice with some of the beet mixture, then grate some cheese over each piece. Serve immediately.
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