Heat broiler to high, with rack set 4 inches from heat. Squeeze juice from grated ginger into a large bowl (you should get about 1 teaspoon juice); discard pulp. Add cream and granulated sugar. Whip until soft peaks form; refrigerate.
Place apricots, cut side up, on a rimmed baking sheet. Dividing evenly, top with brown sugar, butter, and cardamom. Broil until apricots just begin to char, 2 to 5 minutes. Top apricots with whipped cream, and serve immediately.