Preheat oven to 350° F.
Peel peaches (if desired) and then slice into ½-inch thick slices.
In a large bowl, stir together the peaches, ¼ cup granulated sugar, lemon juice, cornstarch, vanilla and ½ teaspoon cinnamon.
Pour the peach mixture into an 8x8-inch square baking dish. Bake the filling for 10 minutes and then remove from the oven.
Meanwhile, mix up the cobbler topping. In a large bowl, whisk together the flour, ¼ cup granulated sugar, baking powder, cinnamon and salt until well combined.
Add the cold butter cubes and use a pastry blender or your fingers to work the butter into the flour mixture until the largest pieces of butter are the size of small peas.
Add the milk and use a fork to mix the dough until it comes together. The dough will be somewhat dry, but should come together after some mixing.
Divide the cobbler dough into 9 portions. Flatten each dough piece slightly so that it is about ¾-inch thick and place on top of the peach filling. The dough pieces do not need to be perfectly round; the cobbler will have a somewhat rustic look to it. Sprinkle the top with sugar, if desired, and use oven mitts to return the hot pan to the oven.
Bake for 40-45 minutes, until the topping is golden brown and cooked through. Check that the underside of the topping is done (not gooey) in the center of the pan by gently lifting it up to check the underside.
Let cobbler cool for about 15 minutes and then serve warm.