Preheat the oven to 450°F, and thoroughly grease your popover pan. Melt the tablespoon of butter in your bowl and set it aside to cool slightly. Break the eggs into the same bowl (with the butter) and whisk gently. Then whisk in the flour, salt, and milk until just barely blended; a few lumps are fine.
Add in the cheddar cheese and chives and, using a rubber spatula, gently fold the cheese and chives into the batter. Don't over mix.
Fill the cups 2/3 full (3/4 full for large popovers), and bake at 450 degrees for 15 minutes. Then, reduce the oven temp to 350 degrees and bake for an additional 20 minutes. (Do not open the oven! Opening the oven door in the first 20 minutes can cause them to collapse.)
When they are brown and crusty, remove them from the oven, and make a small slit in their sides with a sharp knife. Serve immediately.