Place flour, baking powder, baking soda, salt, cardamom, and sugar into the large bowl with cold butter. Cut butter into dry ingredients until it forms a sand-like consistency.
Add milk, vanilla, and room temperature eggs to the smashed pears and mix together.
Heat a 12" cast iron skillet on medium heat. Melt butter and brown sugar together until they sugar starts to melt. Lay the sliced pairs in a circular patter.
Carefully spoon the cake batter over the caramel and sliced pears. Then bake in a 350° oven for 20-25 minutes.