Preheat the oven to 375°F. Butter a square baking pan (8-inch by 8-inch or 9-inch by 9-inch.)
Whisk together the flour, baking powder, and salt in a mixing bowl and set aside. In a separate bowl, whisk together the eggs and 1 cup of the sugar. To that, add the butter and milk and whisk until mostly even with some small chunks of butter still visible. Pour that into the flour mixture and stir just until there are no dry pockets of flour anywhere. Scrape the batter into the buttered baking pan.
Cut the peaches in half, remove the pits but leave the peels on, and slice into 1/8-inch to 1/4-inch thick slices. Set aside.
In a medium, heavy-bottomed saucepan over medium-high heat, stir together the remaining sugar, whiskey lemon juice, and vanilla extract and bring to a hard boil. Boil for 3 minutes, remove the pan from the heat, add the sliced peaches, and toss gently with a rubber spatula or spoon to coat the fruit with the caramel syrup. Spoon the fruit and syrup evenly over the cobbler batter in the buttered pan.
Bake for 30 minutes, or until the edges are caramelized with bubbled syrup and the inside is fully set, but still moist. Let cool for 10 minutes before scooping from the pan, or cool for 30 minutes before cutting into 9 squares and serving. Serve with a drizzle of cold, heavy cream, whipped cream, or ice cream!