In a large bowl, mix together the flours, baking powder, cheese, salt and basil. Add the pieces of butter, working them into the flour, as you would with pie crust, till the mixture forms even crumbs.
Beat together 1 whole egg, 1 egg yolk, and the buttermilk or yogurt. Stir gently into dry mixture till the whole thing clings together.
Turn dough onto a well-floured surface and pat into a 1/2-inch-thick rectangle. Using a bowl scraper, baker's bench knife, regular knife or rolling pizza wheel, cut rectangle into squares; cut each square in half diagonally, so you have triangular scones. Make them as large or small as you wish.
Transfer scones to lightly greased or parchment-lined baking sheet. Whisk reserved egg white vigorously, till it becomes somewhat liquid (instead of "clumpy"). Brush each scone with egg white, and sprinkle with some additional Parmesan.
Bake scones in a preheated 450Â¡F oven for 10 minutes, or until light golden brown. Remove from oven and cool on a wire rack.