Peel, core, and slice the pineapple into ½-inch thick rings. You'll need 6-7 slices. Reserve two tablespoons of the pineapple juice from the cutting board or, using a citrus press, squeeze juice from an extra slice.
Add the butter, light brown sugar, salt, and pineapple juice to a skillet. Cook it over medium heat while stirring occasionally, or until it thickens slightly.
Add the pineapple slices to the brown sugar mixture and cook for 10 minutes, flipping the pineapple halfway through. Take the skillet off the heat.
In a bowl, whisk together the gluten-free flour, cornstarch, salt, baking soda, baking powder, cinnamon, cardamom, and granulated sugar until combined.
Add the softened butter to the dry ingredients and, using your fingers or a pastry cutter, work the butter into the flour mixture until the mixture resembles coarse sand.
In a separate bowl, whisk the eggs, sour cream, and vanilla together until combined. Add the wet mixture to the dry ingredient mixture and whisk to combine.
While continuing to mix the batter, slowly pour in the milk until smooth and pourable. Cover the mixing bowl and let the batter rest for 30 minutes.
While the batter rests, preheat the oven to 350°F (177°C) and grease the sides of a 9-inch round pan.
Arrange the pineapple slices in the bottom of the pan. Pour all of the syrup on top.
Carefully pour the cake batter over the pineapple, making sure not to move around the pineapple as you do. Smooth out the surface and bake the cake for 40-43 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool for about 30 minutes before flipping it over onto a plate.