Heat oven to 350 degrees.
Place a 10-inch round cake dish on top of parchment paper.
Cut a round from the parchment paper that will fit the cake dish.
Butter the 10-inch round cake dish.
Then butter the parchment paper as well.
Pit the peaches and cut into slices about 1/8 to 1/4-inch thick.
Arrange the slices in an overlapping pattern on the bottom of the cake dish.
Combine 3/4 cup sugar, vanilla and ½ cup water in a small saucepan.
Cook over medium-high heat until the mixture turns amber, about 15 minutes.
(Do not stir while the mixture is cooking but you may have to remove from heat occasionally and swirl the mixture).
Remove from heat immediately.
Ladle caramelized sugar evenly over the peaches in the cake dish.
In a medium bowl, combine flour, baking powder and nutmeg.
Set aside.
In another bowl, beat together the butter and remaining 3/4 cup sugar and eggs.
Stir the flour mixture into the egg mixture.
Spread the batter evenly over the peaches and caramelized sugar sauce.
Bake for 30 to 45 minutes, or until cake is set and a toothpick inserted in center comes out clean.
Remove cake from oven and set on a cooling rack.
Run a knife around the sides to loosen edges.
Place a platter or large plate on top of the cake.
Invert the cake onto the platter.
If any of the peach slices stick to the pan, lift them off carefully.
Then replace them on top of the cake. This should not be refrigerated before serving.