Heat the oven to 350 F and grease a 9-inch round cake pan. Melt the butter for the topping with the brown sugar and salt, stirring until smooth, then spread it evenly in the pan.
Arrange the pear slices in overlapping circles over the caramel mixture. Set the prepared pan aside.
Whisk together the all-purpose flour, almond flour, baking powder, baking soda, cinnamon, cardamom and salt.
Beat the softened butter and granulated sugar until light and fluffy. Beat in the eggs one at a time, followed by the sour cream and vanilla.
Add the dry ingredients in three additions, alternating with the milk, and mix just until combined. Spoon the batter gently over the pears and smooth the top.
Bake for 42 to 48 minutes, until the center springs back when lightly pressed. Cool for exactly 15 minutes, loosen the edge and carefully invert the warm cake onto a serving plate.