Preheat oven to 400º F.
In a large bowl mix together the flour, sugar, baking powder, baking soda, and salt until completely combined. Using a pastry cutter or a food processor, mix the butter into the dry mixture until coarse crumbs form. Do not over mix.
Gradually pour in milk or buttermilk and mix until just combined.
Lightly flour a clean surface with all-purpose flour. Pour the dough onto the floured surface and bring the dough together with floured hands. Gently work the dough until it completely comes together. If the dough gets too soft, pop it in the refrigerator for 10 minutes.
Dust a rolling pin with flour and gently roll the dough into a rectangle. Fold one side of the dough into the center, then the other. Roll again until the dough is ¾-inch thick. Fold the dough once more into the center from one side, then the other. Roll again into a ¾-inch thick rectangle.
Using a 2 ¾-inch or 3-inch biscuit cutter, cut 10-12 biscuits from the dough.
Arrange the biscuits on a sheet pan, making sure each biscuit is touching the other to help them rise taller and evenly. Lightly brush with heavy cream and sprinkle with turbinado sugar.
Bake for 18-22 minutes or until golden brown.
While the biscuits are baking, slice the peaches and place into a bowl. Sprinkle granulated sugar over the sliced peaches and gently toss. Refrigerate and set aside until ready to use.
Make homemade whipped cream: Whisk heavy cream, vanilla extract, and granulated sugar together until stiff peaks form. Chill until ready to use.
After the shortcakes cool completely, it's time to assemble. Slice the shortcakes in half, spoon a generous dollop of whipped cream on the bottom shortcake, then follow it with a few peach slices. Place the top shortcake over the peaches, then layer another spoonful of whipped cream followed by more sliced peaches. Serve immediately and enjoy!