Preheat the oven to 425 degrees F (220 degrees C).

Whisk flour, baking powder, and salt together in a bowl. Cut in shortening until mixture resembles coarse crumbs. Stir in Cheddar and chives; gradually stir in milk until a soft dough forms.

Turn dough out onto a lightly floured surface and knead until smooth, about 15 times. Roll dough out to 1-inch thick and cut biscuits using a 2.5-inch cookie cutter. Combine scraps, roll out again, and cut remaining dough into biscuits.