Heat oven to 450°F. Line a baking sheet with parchment.
Whisk flour, baking powder, baking soda, salt, and sugar in a large bowl.
Cut in the cold butter with a pastry cutter until pea-sized bits remain.
Stir in cheddar and chives until evenly distributed.
Make a well and pour in cold buttermilk. Mix gently until a shaggy dough forms.
Turn dough onto a lightly floured surface. Press together and fold 4–5 times to create layers.
Pat to about 3/4-inch thickness. Cut rounds with a 2 to 2 1/2-inch cutter, re-rolling scraps as needed.
Arrange biscuits close together on the sheet. Chill 5 minutes if the dough softened.
Bake until risen and golden, 12–15 minutes.
Brush hot biscuits with melted butter. Cool 5 minutes and serve warm.