Make the topping: position oven rack in center of oven; preheat oven to 350°.
Grease the bottom and sides of a 10 x 3 inch round cake pan (do not use a springform pan, or the caramel may leak out during baking); line the bottom of the pan with a round of parchment paper and grease the paper.
In a saucepan, combine the sugar and water; cook over medium heat, stirring until the sugar dissolves.
Stop stirring; increase heat to high, and cook, occasionally brushing down the sides of the pan with a wet pastry brush to prevent crystallization, until the mixture turns into a golden caramel.
Remove the pan from the heat and immediately whisk in the butter, one piece at a time (be careful—the mixture will bubble up furiously).
Carefully pour the hot caramel into the bottom of the prepared pan.
Cut the peaches in half and discard the pits; cut each peach half into 6 wedges.
Arrange a circle of wedges, overlapping them slightly, around the edge of the pan, on top of the caramel.
Arrange another circle of wedges in the center, facing the opposite direction, covering the caramel completely.
Make the cake: sift the flour, baking powder, baking soda, cinnamon, ginger, and salt together in a bowl; whisk to combine; set aside.
In a small bowl, combine the sour cream and vanilla, blend well; set aside.
In the bowl of an electric mixer, using the paddle attachment, beat the butter on med-high speed until creamy, about 1 minute.
Gradually add in the sugar and beat on high speed until the mixture is lightened in texture and color, 2-3 minutes.
Decrease speed to medium and add in the eggs one at a time, beating well after each addition.
On low speed, add in the flour mixture in three additions, alternating with the sour cream mixture in two additions, mixing just until blended.
Spoon batter in large dollops over the peaches, then smooth it into an even layer.
Bake for 45-50 minutes, until the cake is golden brown and springs back when lightly touched.
Set the pan on a wire rack and cool for 10 minutes.
Run a thin-bladed knife around the edge of the pan; using pot holders, very carefully invert cake onto a cake plate or platter.