Preheat your oven to 170 degrees Celsius (338 degrees Fahrenheit).
In a large skillet or saucepan over medium heat, add the sliced peaches, brown sugar, granulated sugar, lemon juice, vanilla paste, salt, cornstarch slurry, fresh nutmeg, and cinnamon.
Cook the peaches, stirring occasionally, until they have softened and the mixture has thickened slightly, about 5-7 minutes. Remove from heat.
If using canned peaches, remove 80% of the liquid and don't cook them; just add all the ingredients mentioned and set aside.
In a separate bowl, combine the self-raising flour, demerara sugar, caster sugar, whole milk, melted unsalted butter, and vanilla paste.
Stir until the batter is well mixed and forms a thick, smooth consistency.
Allow the cobbler topping mixture to rest for 5 minutes.
After resting, pour the cobbler topping mixture over the cooked peaches in a 12cm (4.7 inches) baking dish.
Bake in the preheated oven at 170 degrees Celsius (338 degrees Fahrenheit) for about 25-30 minutes or until the cobbler topping is golden brown and the peach filling is bubbling.
Remove the peach cobbler from the oven and let it cool for a few minutes before serving.
Serve warm with a scoop of vanilla ice cream or a dollop of whipped cream.
