Ready all ingredients and tools before you start as this goes quickly. Pre-heat oven to 400° for wheat flour or 350° for gluten-free flours. Line a rimmed baking sheet with parchment paper.
Place butter in a medium saucepan and melt over low heat. When butter is melted, add the milk, turn heat to high and bring to a boil. When the milk and butter get to a full boil, dump in the flour and salt all at once. Turn heat down to medium-low and stir like mad with a wooden spoon. Stir, beat and fold until the dough pulls away from the sides of the pan, 1-2 minutes. Move the dough to a medium bowl and cool 2 minutes, stirring occasionally.
Add eggs, one at a time to the bowl, stirring like mad after each one until completely incorporated. It will look like a slimy mess for the first minute but keep stirring. The dough will come together. Lastly stir in the cheese and chives.
With a #40 disher, piping bag with plain tip or tablespoon, drop portions of pastry onto parchment lined baking sheets. Sprinkle a little extra Parmesan on top if desired. Bake for 20-25 minutes or until golden brown. Check the puffs early and watch them as ovens and timing vary. Allow puffs to stand for a few minutes after baking. Serve immediately or cool completely and place in an airtight container. Puffs can be made a day ahead and served at room temperature or warmed in the oven for a few minutes.