A couple of hours ahead of time, take the butter and eggs out of the fridge and let them come to room temperature. When they're sufficiently warmed, preheat the oven to 350ºF.
Prepare the springform pan: rub the bottom and sides of the pan with butter, then cut parchment paper to fit in the bottom, press it in place, and butter the paper too.
Spoon the flour into a medium bowl. Add in the baking powder, salt, and nutmeg. Stir lightly with a fork, then set the bowl aside.
Put the butter and oil in a large bowl. With a hand mixer on medium, beat them together until fairly uniform—don’t worry if there are some lumps of butter. Pour in the 1 cup of sugar (keep the 1 tablespoon of sugar for later) and beat the mixture for 2 minutes. (Throughout this process, keep scraping down the sides with a spatula.) Add the eggs and beat everything well. Finally, add in the sour cream or yogurt and the vanilla and lemon extracts, and beat all together.
Reduce the mixer's speed to low. Then, working in batches, beat in the flour. When it's all incorporated, set aside the mixer. Working gently, fold in 2/3 of the peaches so they are distributed uniformly. Then scrape the batter into the springform pan. Smooth the top, then spread the rest of the chopped peaches on top and press them in lightly. Sprinkle the last tablespoon of white sugar all over the top.
Slide it into the center of the oven. Set a timer for 50 minutes. (This actually took 55 minutes in our oven.) The peach cake is done when a tester inserted near the center comes out completely clean, and the sides are just starting to pull away from the edge of the pan.
Cool on a rack—after 30 minutes, carefully loosen the springform ring and remove it. When it has completely cooled, carefully transfer to a serving plate, removing the parchment paper with a thin straight spatula, such as an icing spatula.