Preheat broiler to high.
Place 4 pineapple rings on a small baking sheet and sprinkle with brown sugar. Broil 3 to 4 inches from the heat source just until the sugar is melted, about 2 minutes.
Whip cream with an electric mixer until stiff. Gently fold in yogurt and vanilla.
Divide the remaining pineapple rings among 4 plates. Top each ring with 2 tablespoons of the cream, 2 slices of cake, side by side, and another 2 tablespoons cream. Top each stack with a ring of sugared pineapple, sprinkle with coconut and serve.