Heat oven to 400 degrees.
Slice peaches in half and discard the pit. Cut into ½-inch thick slices and place in 1-½-quart baking dish, or 9x9 pan. Pour bourbon over the peaches.
Mix together brown sugar, white sugar, cornstarch, and cinnamon in a small bowl. Sprinkle over the top of the peaches and toss to coat the peaches fully.
Bake at 400 degrees for 10 minutes.
While peaches are in the oven, combine the flour, baking powder, cinnamon and salt in a medium bowl.
Slice butter into small slices/chunks and add to the flour mixture. Cut in with a pastry cutter, or using your hands, until the flour becomes fine crumbs.
Make a well in the middle of the flour and pour in the milk and vanilla. Mix until the flour is fully incorporated with the liquids.
Once the peaches have been removed from the oven, drop large spoonfuls of the batter over the top of the peaches. Sprinkle the whole cobbler with cinnamon sugar.
Reduce oven temperature to 375 degrees. Bake an additional 25-30 minutes, or until golden brown. Cool 5 minutes before serving. Serve with vanilla ice cream.