Heat oven to 350°.
Add cold butter to a 10-inch springform pan. Heat in oven until melted, about 3 minutes. Swirl pan to coat bottom with melted butter. Sprinkle evenly with brown sugar.
Fan slices of pear over sugar, with pointed ends toward center of pan, overlapping slightly. Set aside.
In a small bowl, whisk together flour, ginger, baking powder, salt and cardamom. With a stand mixer, beat softened butter in a large bowl until smooth. Add granulated sugar and beat 2 minutes, until creamy. Beat in eggs, one at a time, beating well after each addition. Beat in vanilla.
On low speed, beat in flour mixture, alternating with milk, beginning and ending with flour mixture. Spoon batter over pears and spread to pan edge with a spatula. Wrap bottom of pan with foil and bake at 350° for 55 to 60 minutes. Test center of cake with a toothpick; if pick tests clean, remove to a wire rack. Cool 5 minutes.
Invert cake onto plate; remove side of pan. Carefully lift off pan bottom and cool cake to room temperature. Serve slightly warm with ice cream, if desired.
