Preheat oven to 450F and line a baking sheet with parchment paper or a silpat liner.
Whisk together the flours, baking powder, sugar, and salt in a large bowl. Cut in the butter with a fork or using two butter knives until it looks like coarse meal. Sprinkle in the dill; stir in the egg and ½ cup heavy cream, adding more cream 1 tablespoon at a time as needed until the dough comes together but isn’t too wet (you shouldn’t need more than 2 additional tablespoons of cream). Use your hands to mix in the cheddar, being careful not to over-process.
Shape the dough into a ball, then flatten it into a disk. Wrap it in plastic wrap and chill 10 minutes in the freezer.
Turn the dough out onto a lightly floured surface. Roll or press the dough out to a circle about 7 to 8 inches in diameter and ¾ inch thick. Use a floured 2 ½ inch round cookie cutter to stamp out the scones, then gather up the dough scraps and repeat as necessary (you should get about 10 scones).
Transfer the scones to the prepared baking sheet, lightly brush the tops with a little milk, and bake until puffed and light golden brown on top and bottom, about 14 to 16 minutes.
Notes
To omit the whole wheat flour, use 1 cup of all-purpose flour instead.