Preheat the oven to 350°F (175°C). In a small saucepan over medium heat, melt the butter. Stir in the brown sugar and cook for about 1 minute, until the sugar dissolves and forms a smooth caramel-like mixture.
Pour this mixture into a 9-inch round cake pan, spreading it evenly over the bottom.
Arrange the pineapple rings on top of the sugar mixture.
In a medium bowl, whisk together the flour, baking powder, cinnamon, and salt.
In a large bowl, using an electric mixer, cream together the softened butter and granulated sugar until light and fluffy (about 2 minutes).
Add the eggs one at a time, mixing well after each addition.
Stir in the vanilla extract.
In a small bowl, mix the sour cream, pineapple juice, and milk.
Gradually add the dry flour mixture and the wet sour cream mixture to the butter mixture, alternating between the two.
Begin and end with the flour mixture, mixing until just combined after each addition.
Carefully spoon the batter over the arranged pineapple in the cake pan, smoothing it out evenly.
Bake for 40–45 minutes, or until a toothpick inserted in the center comes out clean. Remove from the oven and let cool in the pan for 10 minutes. Run a knife around the edges of the pan to loosen the cake. Place a serving plate upside down over the cake pan and carefully invert the cake onto the plate. Let it sit for a few minutes before lifting off the pan, allowing the caramel topping to settle.
