Preheat your oven to 170 degrees Celsius (338 degrees Fahrenheit).
In a large skillet or saucepan over medium heat, combine the sliced peaches, brown sugar, granulated sugar, lemon juice, vanilla paste, salt, cornstarch slurry, fresh nutmeg, and cinnamon.
Cook the mixture, stirring occasionally, until the peaches soften and the mixture thickens slightly, which should take about 5-7 minutes. Remove from heat.
If using canned peaches, drain 80% of the liquid and skip the cooking step; just mix in all the other ingredients and set aside.
While the peach mixture is cooking, prepare the cobbler topping by mixing the self-raising flour, demerara sugar, caster sugar, whole milk, melted unsalted butter, and vanilla paste in a separate bowl.
Stir until the batter is well combined and has a thick, smooth consistency.
Let the cobbler topping mixture rest for 5 minutes to thicken further.
After resting, pour the cobbler topping over the cooked peaches in a 12cm (4.7 inches) baking dish.
Bake in the preheated oven for about 25-30 minutes, or until the cobbler topping is golden brown and the peach filling is bubbling.
Once done, remove the peach cobbler from the oven and allow it to cool for a few minutes before serving.
Serve warm, optionally with a scoop of vanilla ice cream or a dollop of whipped cream for an extra treat.
