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. Stir until the peaches are evenly coated. Pour the peach mixture into a greased 9x13-inch baking dish.

In a separate bowl, mix the flour, rolled oats, salt, and cold cubed butter. Use a pastry cutter or your fingers to work the butter into the dry ingredients until the mixture resembles coarse crumbs. Sprinkle this topping evenly over the peaches.

Place the baking dish in the preheated oven and bake for 35-40 minutes, or until the topping is golden brown and the peach filling is bubbly. Allow the cobbler to cool for a few minutes before serving.