For the cake: Preheat the oven to 350 degrees F.
Put the brown sugar, 1/2 stick of the butter and the 1/4 cup pineapple juice in a small saucepan over medium heat. Cook until the sugar dissolves and begins to caramelize, about 10 minutes. Pour into a 9-by-13-inch baking dish and top with the pineapple rings. Set aside.
Whisk together the flour, ginger and salt in a medium bowl.
In a stand mixer with the paddle attachment, beat together the remaining 2 sticks butter and the granulated sugar until light and fluffy, about 4 minutes. Beat in the eggs, one at a time, scraping down the bowl as needed. Beat in vanilla until well incorporated.
With the mixer on low speed, slowly add in the dry ingredients, beating just until incorporated--be careful not to over-mix. Spread the batter over the pineapple rings in the baking dish.
Bake until a toothpick inserted into the center of the cake comes out clean, about 30 minutes. Cool for 15 minutes.
Meanwhile, make the topping: Stir together the sour cream, sugar and vanilla seeds in a small bowl.
To serve, invert the cake over a platter and cut into portions. Serve with a dollop of the sour cream topping.