Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
Place the flour, baking powder, salt and 2 tablespoons plus 1 1/2 teaspoons sugar in a large mixing bowl. Cut the butter into 1/2-tablespoon-size bits and add to the flour. Using your fingertips or a pastry cutter, work the butter into the flour mixture until it resembles a coarse meal. Make a well in the center of the flour and pour in all the heavy cream, sour cream and 2/3 cup plus 1 tablespoon half-and-half.
Stir the dry ingredients into the cream, then add the chives and mix with a wooden spoon until the dough just begins to come together into a ball. Turn the dough onto a lightly floured surface and knead 2 or 3 times to form a cohesive mass. (Do not overwork the dough.)
Using a rolling pin, roll the dough to a 1-inch thickness. Dip a 2 1/2-inch biscuit cutter in flour, then cut the dough. Repeat until all the dough has been cut. Scraps can be gathered together and re-rolled one more time.
Place the biscuits on the prepared baking sheet, leaving about 1/4-inch between them.
Brush the tops of the biscuits with the remaining tablespoon half-and-half, and sprinkle with the remaining tablespoon sugar.
Bake until lightly browned on top and flaky in the center, about 20 minutes.