Step 1: Preheat your oven to 375°F (190°C). Lightly butter a 9x13 inch baking dish.
Step 2: Prepare the peaches. If using fresh peaches, bring a large pot of water to a boil. Carefully drop peaches into the boiling water for 30-60 seconds, then immediately transfer them to an ice bath. This helps the skins slip right off. Peel, pit, and slice the peaches into roughly 1/2-inch thick wedges. If using frozen peaches, thaw them completely and drain well. You want about 6 cups of sliced peaches.
Step 3: In a large bowl, combine the sliced peaches with granulated sugar, brown sugar, lemon juice, vanilla extract, cinnamon, nutmeg, cornstarch, and a pinch of salt. Toss gently to coat the peaches evenly.
Step 4: Pour the peach mixture into the prepared baking dish and spread it into an even layer. Set aside.
Step 5: For the topping, in a separate medium bowl, whisk together the all-purpose flour, granulated sugar, baking powder, and salt.
Step 6: Add the cold, cubed butter to the dry ingredients. Using a pastry blender, your fingertips, or two knives, cut the butter into the flour mixture until it resembles coarse crumbs with some pea-sized pieces remaining.
Step 7: Pour in the milk and almond extract (if using). Stir with a fork just until the dry ingredients are moistened. Do not overmix.
Step 8: Drop spoonfuls of the topping mixture evenly over the peach filling. Don't worry if it doesn't cover every single peach; it will spread as it bakes. Optionally, sprinkle turbinado sugar over the topping for a crunchy crust.
Step 9: Bake for 45-50 minutes, or until the topping is golden brown and the peach filling is bubbling and thickened. If the topping browns too quickly, loosely tent the dish with foil.
Step 10: Remove from the oven and let the peach cobbler cool for at least 15-20 minutes before serving. This allows the filling to set.