Preheat oven to 350 degrees. Lightly spray a 9-inch round cake pan.
In a small skillet, melt 2 tablespoons butter over medium heat. Add the brown sugar and 1/4 teaspoon cardamom, stirring constantly. When the sugar dissolves, pour it into the pan. Arrange the sliced pears on top of the syrup in an overlapping circle.
Sift together flour, baking powder, cardamom, and salt, stir to combine. In a separate bowl, blend butter and granulated sugar, then blend in eggs. Alternate dry ingredients with milk, blending to combine. Add vanilla.
Spoon batter into pan, starting in center and gently spreading on top of the fruit.
Bake for 50 minutes, or until done.
Cool in pan 10 minutes, then cover with a plate and invert it onto the plate. Leave the pan on for two minutes, then gently remove the pan.