Preheat the oven to 350ºF (180ºC.) Melt the butter in a skillet, watching it carefully and swirling it gently. When it starts to take on a light brown color and smells a little nutty, and the bubbles begin to subside, pour the browned butter into an 8-inch (20cm) square baking pan, leaving the blackened bits back in the skillet.
In a medium bowl mix together the flour, sugar, baking powder, lemon zest, and salt with a whisk. Add the milk and vanilla extract, and stir until smooth.
Pour the milk mixture over the butter in the pan without stirring. (Don't worry, the batter will rise up over the peaches as they are baking.) Toss the peach slices with the cinnamon and strew them over the batter in an even layer. Crumble the brown sugar over the top and baking until the top is golden brown and the center of the cobbler feels just set, about 40 minutes. Remove from the oven and cool on a wire rack.