Slice kumquats into 1/4 inch slices, removing the seeds as you go.
Melt 6 tbsp of the butter and add it to the bottom a 10-in baking pan with the brown sugar and mix well. Arrange kumquat slices in an even layer over the mixture.
Whisk the flour, baking powder and salt in a medium bowl and set aside.
Using an electric mixer on medium speed, beat remaining cup of butter with sugar until fluffy, about 5 minutes. Add egg and vanilla and beat until combined.
Alternately add flour mixture and milk with the mixer on low, beating well after each addition. Use a rubber spatula to scrape down sides of the bowl as needed.
Pour the cake batter over the kumquat slices (batter will be thick) and bake for 35-45 minutes or until a toothpick inserted comes out clean.
After removing from the oven, immediately invert onto a plate.