Preheat the oven to 400º. Either line a large baking sheet with parchment paper, or gather a large cast iron skillet (or a couple of smaller ones).
Stir the milk and vinegar together in a small container, such as a liquid measuring cup. Place the container in the fridge to keep the mixture cool while you prepare the dry ingredients.
Whisk the flour, sugar, baking powder, baking soda and salt together in a large mixing bowl.
Cut the butter into eight to ten slices and add them to the bowl. Use a pastry cutter, fork, or a couple of butter knives to cut the butter into the dry ingredients. Continue until the mixture resembles coarse crumbs.
Remove the milk mixture from the fridge and pour it into the bowl with the dry ingredients. Stir everything together just until the dough is combined. Don't overwork the dough. It will appear uneven and shaggy, which is fine.
Turn the dough out onto a lightly floured surface. Use an approximately 3-inch round biscuit cutter (Note 3) to cut circles in the dough. Dip the cutting edge in flour before making each cut, and avoid twisting it when cutting the dough circles.
Gently re-roll any leftover dough and continue making circles until all the dough is used.
Place each dough round into the skillet(s) or on the baking sheet, arranging them so that the edges are just touching.
Lightly brush the tops of the biscuits with non-dairy milk.
Bake the biscuits for 15 to 18 minutes, until they're fluffy and golden brown.
Place the skillet or baking sheet on a cooling rack and let the biscuits cool for about 10 minutes. Remove them from the skillet or baking sheet and place them directly on the racks to finish cooling.