Preheat the oven to 350°F, and prepare a 9-inch cake pan.
Add the butter to a skillet, and melt over a medium heat. Turn it down if it begins to brown.
Add the sliced peaches, and cook, stirring occasionally, for about 8 to 10 minutes over a low setting. The peaches should be softened, but not mushy. Strain the butter, and set it aside.
Measure the flour and baking powder into a bowl, and toss to combine.
In a second bowl, whisk the eggs and egg yolk together until they are thoroughly mixed but not too frothy.
Add the sugar, vanilla extract, lemon zest, and strained butter to the eggs, whisking together thoroughly.
Add the flour mixture to the wet ingredients by the spoonful, stirring slowly but thoroughly with each addition.
Add the cooked peaches and mix them into the batter.
Spoon or ladle the cake batter into the pan and bake for about 30 minutes. The top of the cake should be a golden brown color.
Remove from the oven and allow it to cool for about a half an hour. Once cooled, flip it out of the pan, and dust with powdered sugar.