Preheat the oven to 350℉, then lightly butter a metal 9x9 baking dish and set aside—you can also use an 8x8, but it will take longer to cook.
Prepare the filling. Place the peeled, pitted, and sliced peaches into a large bowl with the remaining filling ingredients: the sugar, brown sugar, flour, lemon juice, vanilla, salt, and spices. Stir until well combined, then pour the peach filling into the prepared dish.
Prepare the cobbler topping. In another large bowl, mix together the flour, sugar, baking powder, and salt.
Using a fork or your fingertips, pinch the butter into the dry ingredients until a coarse meal forms with pea-sized clumps. Toss in the oats and the toasted pecans.
Pour in the milk or cream, then fold everything together with a rubber spatula until the dry ingredients are well hydrated and a thick batter forms. Evenly dot the cobbler topping over the peaches with a spoon or cookie scoop, then sprinkle over the demerara sugar (if using).
Bake the cobbler for 40 minutes, until the top is golden brown and the filling is bubbling. A toothpick inserted into the topping should come out clean.
Allow the cobbler to cool for 15-20 minutes before serving with vanilla ice cream.