Preheat oven to 350°F (180°C). Butter and flour a 9-inch round cake pan.
In the bowl of a stand mixer fitted with the paddle attachment, whisk together flour, granulated sugar, nutmeg, and salt by hand. Add eggs, buttermilk, and vanilla, and beat at medium-low speed until smooth. Add ½ cup (113 grams) butter, beating to combine. Set aside.
Place remaining ¼ cup (57 grams) butter in prepared pan. Place pan in oven until butter is melted, 2 to 3 minutes. Sprinkle brown sugar over melted butter. Top with peaches, arranging as desired in a single layer. Spoon batter over peaches, smoothing top with an offset spatula.
Bake until a wooden pick inserted in center comes out clean, 40 to 50 minutes, covering with foil after 30 minutes of baking to prevent excess browning. Run a knife around edges of pan to loosen cake. Invert cake onto a serving plate, and let cool slightly. Serve warm.
Recipe by Bake from Scratch at https://staging.bakefromscratch.com/peach-upside-down-cake/