Preheat oven to 425F. Line a baking sheet with parchment paper. Set aside.
In a medium bowl, whisk together well the flour, sugar, baking powder and salt. Set aside.
Grate cheddar cheese and dice chives. Add to flour mixture and toss with your fingers to coat/separate/combine the cheese and chives with the flour mixture. Add cold butter chunks to flour mixture. Using your fingertips, rub the butter in to the flour mixture until you have coarse crumbs with butter pieces no larger than a pea. Add cold buttermilk and using a fork, stir in to mixture until flour is evenly moistened. YOU MAY NEED TO ADD A TEASPOON OR TWO ADDITIONAL BUTTERMILK, to thoroughly moisten.
Turn dough out on to floured surface. Gather the dough in to a mound and fold it over on to itself 2 or 3 times, until dough is one moist piece. Form in to a ball. Using a floured hand, press dough down until you have an even 1-inch thick piece of dough. Using a 3-inch cutter, cut rounds from the dough and place on to parchment lined baking sheet, allowing several inches between each biscuit. Take scraps of dough and re-form in to a 1-inch thick piece and cut more. You should get 6 or 7 biscuits. Brush tops of biscuits with a bit of milk or cream.
Bake in preheated oven for 15-17 minutes or until lightly golden and just cooked through. Remove from oven, then remove biscuits from baking sheet on to a cooling rack. Reduce oven temperature to 350F and reserve baking sheet with parchment.
Allow biscuits to cool for 15 minutes (or if making right ahead, allow to cool completely then store in airtight container until ready to make).
With a sharp knife, cut around the top of the biscuits, cutting straight down, about 1/2-inch in from the outside edge and about 1 inch deep. Once you have cut a circle all the way around, use the tip of the knife to lift out the round. Pull out bits of biscuit as needed to have a 1-inch deep hole that is flat on the bottom. Place biscuits back on to baking sheet with parchment, allowing several inches between each one. Break an egg in to the hole in each biscuit. Season egg with a bit of salt and pepper.
Return to the 350F oven and bake for 20-25 minutes, depending on how runny you like your eggs. Check at 18-20 minutes and if biscuits are getting too brown, you can loosely cover the top with a sheet of aluminum foil. Once the egg whites are set and the egg yolk is set slightly, remove from oven. Allow to cool at least 5 minutes before eating, to allow egg to set up a bit. Garnish with additional chopped chives and freshly ground pepper. These are good warm or at room temperature. Refrigerate left-overs.
*Make your own buttermilk by adding 2 tsp vinegar to a 3/4 cup milk and allow to stand 5 minutes.