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. Let the mixture sit for about 10 minutes to allow the sugars to dissolve and the peaches to release their juices.

In a separate bowl, whisk together the flour, baking powder, and salt. Add the chilled butter and vegetable shortening, and use a pastry cutter or your fingers to work the fat into the flour mixture until it resembles coarse crumbs. Gradually add the milk, stirring just until the dough comes together. Be careful not to overmix.

Transfer the peach mixture to a 9x13-inch baking dish, spreading it out evenly. Drop spoonfuls of the biscuit dough over the peaches, leaving some gaps for the peaches to bubble through. The dough does not need to cover the entire surface.

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

Allow the cobbler to cool for about 10 minutes before serving. This dessert is best enjoyed warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.