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 10 minutes to allow the juices to form.

In a separate bowl, mix the flour, rolled oats, baking powder, and salt. Add the cold, cubed butter and use a pastry cutter or your fingers to work the butter into the dry ingredients until the mixture resembles coarse crumbs. Drizzle in the honey or maple syrup and mix until combined.

Pour the peach mixture into a greased 9x13-inch baking dish, spreading it out evenly. Sprinkle the cobbler topping evenly over the peaches, ensuring they are fully covered.

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. If the topping begins to brown too quickly, cover the dish with aluminum foil and continue baking.

Remove the cobbler from the oven and let it cool for 10 minutes before serving. This dessert is best enjoyed warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.