Heat oven to 375°F. Place butter in a 9×13-inch dish and melt in the oven.
Toss peaches with 2/3 cup sugar, lemon juice, cinnamon, nutmeg, and cornstarch. Let sit 5 minutes.
Whisk flour, 3/4 cup sugar, baking powder, and salt. Stir in milk and vanilla until smooth.
Remove hot dish from the oven. Swirl butter to coat, then pour in the batter without stirring.
Spoon peaches and their juices evenly over the batter. Do not stir.
Bake 35–45 minutes, until the top is deeply golden and the edges bubble.
Cool 10 minutes. Serve warm, ideally with vanilla ice cream or whipped cream.