Step 1: Preheat your oven to 375°F (190°C). Butter a 9x13 inch baking dish.
Step 2: Peel, pit, and slice the fresh peaches into roughly 1/2-inch thick pieces. In a large bowl, gently toss the sliced peaches with 1/2 cup granulated sugar, 2 tablespoons all-purpose flour, 1 teaspoon vanilla extract, and 1/2 teaspoon ground cinnamon.
Step 3: Pour the peach mixture evenly into your prepared baking dish.
Step 4: In a separate medium bowl, whisk together 1 1/2 cups all-purpose flour, 1/2 cup granulated sugar, 1 tablespoon baking powder, and 1/4 teaspoon salt.
Step 5: Add 1/2 cup (1 stick) cold, unsalted butter, cut into small cubes. Using a pastry blender, two forks, or your fingertips, cut the butter into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 6: Pour in 1/2 cup milk and stir with a spoon until just combined. Do not overmix.
Step 7: Drop spoonfuls of the topping mixture over the peaches in the baking dish. You don't need to spread it perfectly. Sprinkle the top with 1 tablespoon of granulated sugar.
Step 8: Bake for 40-50 minutes, or until the topping is golden brown and the peach filling is bubbly and tender. Let the cobbler cool for at least 15-20 minutes before serving.