Think of this as "new and improved" borscht, with fruit and mint added for freshness and subtle spices for intrigue. If you like, roast extra beets to eat later in the week; they keep several days in the refrigerator.
Preheat oven to 400°. Wrap each beet in foil and set in a pie pan. Roast beets until easily pierced with a fork, 1 1/2 to 2 hours. Cool, then peel and discard skins.
Pit plums over a food processor to catch any juices. Roughly chop plums and beets and put in food processor with 2/3 cup ice water, 2/3 cup onion, 1 1/2 tbsp. mint, the vinegar, 1 tsp. salt, and 3/4 tsp. baharat. Drape processor with paper towels to catch any spills. Whirl soup until smooth.
Transfer soup to a bowl, stir in 1 cup more ice water, and chill, covered, at least 2 hours.
In a small bowl, whisk yogurt, honey, and remaining 1/4 tsp. baharat. Season soup to taste with more salt if you like. Ladle into bowls, add a spoonful of spiced yogurt to each, and swirl. Scatter remaining mint on top and add a little more onion if you like.
*Find baharat, a Middle Eastern blend of black pepper and spices, at well-stocked grocery stores. Or combine 1 tsp. each paprika, pepper, cinnamon, coriander, and cumin; 1/2 tsp. each ground allspice and cardamom; and 1/4 tsp. each ground cloves and nutmeg. Keeps, airtight, up to 6 months.
Make ahead: Through step 3, up to 1 day, chilled.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.