To make the peach filling: Place the sliced peaches in a large bowl. Set aside.
In a small bowl, combine the granulated sugar and lemon zest. Rub the mixture together with your fingers until fragrant. Add the lemon sugar to the peaches along with the brown sugar, cornstarch, and cinnamon. Gently stir until the peaches are well coated. Let the mixture sit while you prepare the cobbler topping.
To make the cobbler topping: In a medium bowl, whisk together the flour, sugar, baking powder, and salt. Cut the cold butter into the flour mixture with a pastry blender or two forks, or by using clean hands and pinching the butter between your forefinger and thumb. Mix until the butter pieces are pea-sized.
In a medium bowl, whisk together the cold buttermilk and vanilla. Pour the mixture into the dry ingredients and stir with a spatula until the dough just comes together. It will be sticky; don’t over mix.
Pour the peach mixture into a 9x13-inch pan. Drop pieces of the cobbler topping on top of the peaches—it is fine if there is space between the dollops. Lightly brush the cobbler topping with the heavy cream and sprinkle with the turbinado sugar.
Bake for 45 to 50 minutes, until the cobbler topping is golden brown, and the peaches are bubbling. Let cool on a wire rack for 15 minutes before serving with vanilla ice cream, if desired.
Store cooled leftover cobbler covered in the fridge for up to 3 days.