Preheat the oven to 450ºF. Line two large baking sheets with parchment paper.
Whisk together flour, whole wheat flour, sugar, baking powder, and cream of tartar in a large bowl. Cut in the butter using a pastry cutter or by rubbing the pieces of butter quickly between your fingers until the pieces are the size of peas.
In a small bowl whisk together egg, milk, crushed garlic, and shredded cheese. Add to the flour-butter mixture and mix with a wooden spoon until the mixture just comes together.
Turn onto a floured counter and knead 10-15 times until mixture comes together into a ball. Roll out to ½ inch-thickness cut with a 2-inch biscuit cutter or a floured glass.
Place the biscuits on the prepared pans (12 per pan). Bake for 10-15 minutes or until golden brown.