Preheat oven to 350ºF.
Gently rub the peaches under running water to remove fuzz, and slice peaches. (You will have about 4 cups.)
Place butter in an 11- x 7-inch baking dish; place dish in oven until butter melts.
Stir together flour, sugar, milk, and vanilla in a bowl. Pour onto melted butter. (Do not stir.) Spoon peaches over mixture. (Do not stir.)
Bake in preheated oven until brown and bubbly, 45 to 50 minutes.
