Preheat the oven to 325 degrees F (165 degrees C). Grease the bottom and sides of two 9-inch round, 1 1/2-inch tall cake pans.
Combine brown sugar and butter for topping in a saucepan over medium heat; stir until sugar dissolves and mixture is bubbly, about 3 to 4 minutes.
Pour mixture into one of the prepared cake pans. Arrange pineapple rings in a single layer on top.
Whisk together both flours, baking powder, and salt for cake in a large mixing bowl. Beat sugar and butter in a separate bowl with an electric mixer until creamy and light. Add eggs, one at a time, beating well after each addition. Beat in vanilla. Add dry ingredients alternately with sour cream in two additions each, beating well after each addition.
Pour 1/2 of the cake batter over the caramel and pineapple topping. Pour remaining cake batter into the second cake pan.
Bake in the preheated oven until a toothpick inserted in the center of each cake comes out clean, 60 to 75 minutes.
Remove from the oven and let sit in the pans for 5 minutes. Turn the cake without the topping layer onto a serving platter. Turn the cake with the topping layer on top of that. Serve warm.