Preheat the oven to 350 degrees F.
Place the butter (as is–you will melt this soon) in an 8-inch square or similar size baking dish. Set aside.
Combine the flour, sugar, baking powder, and salt in a medium bowl. Add the milk and vanilla, stirring just until the dry ingredients are thoroughly combined. As you are finishing up with the batter, place the baking dish in the oven to melt the butter. (You don’t want the butter to burn, so avoid putting the dish in the oven too early.It should take about 4-5 minutes to melt cold butter.)
Once the butter has fully melted, remove the baking dish from the oven, and pour the batter over the melted butter. Do not stir.
Evenly distribute the chopped peaches over the batter. Again, do not stir. Sprinkle with a little cinnamon sugar, if desired. (I use about 1 1/2 teaspoons of sugar and 1/8 teaspoon cinnamon, but you can eyeball it or skip it entirely.)
Bake the cobbler for 35 to 45 minutes. The batter will rise up and around the peaches as it cooks. You’ll know the cobbler is cooked throughout when the edges are a deep golden brown. Check a few minutes early and adjust the time if necessary, as all ovens vary.
Serve the cobbler warm or at room temperature, with a scoop of optional ice cream or whipped cream.To best preserve freshness, cover and refrigerate any leftovers.