In the bowl of your stand mixer, combine the chilled butter with the sugar, salt and flour. Cut the butter into those dry ingredients until the mixture looks like shaggy crumbles.
Add the baking powder, baking soda, Hatch chiles and cheese and sourdough starter. Mix briefly, just until combined. Dough should have come together nicely, with just a few crumbles at the bottom of the bowl.
Transfer dough to a well-floured surface. Pat out with your hands until 1/2" thick. Shape isn't too important at this point.
Fold into thirds as you would a letter. Pat out dough again to 1/2 inch thick, and then turn the dough 90 degrees. Fold into thirds again, as you would write a letter to create a rather rough rectangle. Push to 1/2 inch thick, and then repeat: turn, fold, and pat out - this will help create the flaky layers of the biscuits as they bake.
After repeating the folding and patting 3-4 times, the dough should feel light and airy, like a pillow. Push out to 1/2 inch thick. Use the mouth of a canning jar or a biscuit cutter to cut round biscuits - depending on the size of your jar/biscuit cutter, you should get 6-8 biscuits.
Transfer biscuits to a lined baking sheet, spacing them 1 inch apart. Gently take the scraps, and pat them out into a rectangle, and cut biscuits from the remaining dough.
Brush the tops of your biscuits with belted butter, and bake at 425 degrees F for 10-12 minutes or until lightly browned. Serve with butter.
Want to make these without the Hatch chiles? You can simply omit the chiles and the cheese from the biscuit recipe and proceed with the ingredients remaining.