Preheat oven to 450F and line a baking sheet with parchment paper or a silpat liner.
Whisk together the flour, baking powder, salt, and thyme in a large bowl.
Cut in the butter with a fork or using two butter knives until the butter is mostly incorporated, but there are a few larger pieces about the size of peas. Stir in ⅔ of the shredded cheese until it’s completely coated with flour (the ⅓ of the shredded cheese for topping).
Mix in the scallion. Use a fork to mix in the cream and then the milk until the dough comes together; knead it a couple times with your hands. (The dough should come together, but not be too wet. If the dough is still too crumbly, you can add milk a little at a time until it comes together.)
Shape the dough into a ball, then flatten it into a disk; wrap it in plastic wrap and chill 10 minutes in the freezer. (Or chill it in the fridge a couple hours or overnight.)
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 8 to 10 scones).
Transfer the scones to the prepared baking sheet, lightly brush the tops with a little milk and sprinkle on the remaining ⅓ of the shredded cheese.
Bake until the scones are puffed and light golden brown on top and bottom, about 14 to 16 minutes.
Notes
To make ahead, prepare the dough, wrap it in plastic wrap, and store it in the fridge for up to 3 days before rolling out the dough and baking the scones. To make the dough easier to roll out, let it sit for 10 minutes at room temperature first.