In a small bowl, stir together the sugar and ginger.
Line a baking sheet with parchment or foil. Set the apricot halves cut sides up in a single layer. Sprinkle the ginger-sugar mixture evenly over the hollows and tops of all the apricots. Set a broiler rack 5 inches from the heat source. Broil the apricots for 4 to 5 minutes, watching carefully. The sugar should melt, then bubble, then begin to burn along the edges. (If having some black spots bothers you, pull the apricots out of the broiler a bit earlier.) Cool for a few minutes and then serve warm, or squeeze on a touch of lemon juice (to inhibit browning), pack into a quart-size jar, cover, and refrigerate until cold before serving. (They will keep, covered, for up to 1 week.)
To serve, spoon the yogurt into serving bowls. Top with the roasted apricots.
Tip: Once the leftover apricots are refrigerated, the sugar topping will liquefy and bathe the apricots in syrup. You can stir a bit of this syrup into your yogurt, if desired, or simply lift out the apricots with a slotted spoon. You can also puree the apricots, creating a smooth, velvety sauce.