Preheat your oven to 375°F (190°C). In a large bowl, combine the sliced peaches, granulated sugar, brown sugar, cinnamon, nutmeg, vanilla extract, and lemon juice. Toss gently to coat the peaches evenly. Allow the mixture to sit for 10 minutes to let the juices develop.

In a separate bowl, whisk together the flour, baking powder, and salt. Add the chilled butter cubes and use a pastry cutter or your fingers to work the butter into the flour mixture until it resembles coarse crumbs. In a small bowl, whisk together the milk and egg, then add this mixture to the flour mixture. Stir until just combined to form a soft dough.

Transfer the peach mixture to a greased 9x13-inch baking dish. Drop spoonfuls of the dough evenly over the peaches. The dough will spread as it bakes, so don't worry about covering every inch.

Place the baking dish in the preheated oven and bake for 40-45 minutes, or until the crust is golden brown and the peach filling is bubbly. If the crust begins to brown too quickly, tent the dish with aluminum foil.

Allow the cobbler to cool for a few minutes before serving. This dessert is best enjoyed warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.