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.