Preheat the oven to 350°F and prepare the peaches. You can peel the peaches with a vegetable peeler or see the notes for a quick trick. Cut the peaches into 1/3-inch-wide slices.
Add all the filling ingredients into a small bowl: the peaches, lemon juice, lemon zest, vanilla extract, light brown sugar, cornstarch, ground cinnamon, ground ginger, ground nutmeg, and salt. Mix well until the peaches are fully coated. Set aside.
Place the Dutch oven on the stove over medium heat. Add the pieces of butter and melt them into brown butter. Occasionally, stir the butter to prevent it from burning for about 5 minutes or until it turns golden brown. Some milk solids at the bottom of the pan will be brown, and that's what you want! Immediately turn the heat off. The brown butter adds a nutty and depth of flavor to the cobbler.
Add the all-purpose flour, sugar, baking powder, and salt into a mixing bowl. Combine the ingredients with a paddle attachment.
Whisk together the egg, milk, and vanilla extract. While on low speed, slowly pour the mixture into the dry ingredients until the batter comes together. Do not overmix!
Pour the batter into the Dutch oven with the brown butter. Do not stir! Distribute the peaches (with any excess juice) evenly over the batter. Do not stir!
Bake for 50 - 60 minutes until the peach cobbler is golden brown and baked through (when a cake tester or toothpick comes out clean from the topping, not the peach filling). The batter should rise above the peaches.
Serve warm with vanilla ice cream, and enjoy!