Heat the oven to 375°F and toss the peaches with 1/4 cup sugar, lemon juice, vanilla and cinnamon.
Melt the butter in a 9-by-13-inch metal baking pan in the oven until it foams and shows brown flecks, 5 to 7 minutes; watch it closely.
Whisk the flour, remaining 1/2 cup sugar, baking powder and salt in a bowl, then whisk in the milk until mostly smooth.
Pour the batter evenly over the brown butter without stirring them together.
Spoon the peaches and their juices over the batter, again without stirring.
Bake until the edges are crisp, the top is deep gold and no wet batter remains in the center, 45 to 50 minutes.
Rest for 20 minutes so the fruit juices thicken before spooning into bowls.