Preheat your oven to 450°F (230°C). Line a baking sheet with parchment paper.
Place your butter and shortening in the freezer for 15 minutes before starting. Chill your mixing bowl as well if your kitchen is warm.
In your chilled bowl, whisk together the flour, baking powder, baking soda, and salt until well combined.
Using a pastry cutter or two knives, cut the cold butter and shortening into the flour mixture until it resembles coarse crumbs with some pea-sized pieces remaining.
Add the cold sour cream to the mixture and stir gently with a wooden spoon or rubber spatula. Add just enough cold milk to bring the dough together without making it wet or sticky.
Turn the dough onto a lightly floured surface. Instead of kneading, use a folding technique: flatten the dough into a rectangle, fold it in thirds like a letter, then give it a quarter turn. Repeat this process 4-5 times.
Pat the dough to about 3/4 inch thickness. Using a 2.5-inch biscuit cutter, press straight down without twisting to cut out biscuits.
Place biscuits on the prepared baking sheet with sides touching for higher rising (or space them apart for crispier sides).
Let the biscuits rest for 10 minutes to allow the gluten to relax.
Brush the tops with melted butter and bake for 12-15 minutes until the tops are golden brown.
Brush once more with butter when they emerge from the oven. Serve warm.