Toss peaches with both sugars, cinnamon, nutmeg, lemon juice, and vanilla. Set aside for 10 minutes to macerate.
Place butter in a 9×13-inch baking dish and slide into the oven as it preheats to 375°F. Remove as soon as butter is fully melted, about 3–4 minutes.
Whisk together flour, sugar, baking powder, and salt. Add milk and vanilla; whisk until smooth.
Pour batter directly over the melted butter. Do NOT stir. Spoon peaches and all their juice over the batter. Do NOT stir. Mix the cinnamon-sugar and sprinkle across the top.
Bake at 375°F for 40–45 minutes until the top is deep golden, the edges are bubbling, and the batter has risen up through the peaches.
Rest 10–15 minutes before serving. Serve warm with vanilla ice cream.