Preheat the oven to 425 degrees F.
Whisk together the flour, baking powder and salt in a large bowl. Stir in the cheese. Cut the butter into the flour mixture with a pastry blender or 2 knives scissors-fashion until it resembles coarse crumbs. Gently mix in the cream until combined.
Pat the dough out to a 9-inch-thick round on a lightly-floured work surface. Cut out 16 rounds using a 2 1/2-inch biscuit cutter or the rim of a glass, re-rolling any scraps (don't twist the cutter as you use it; this will prevent the biscuits from rising unevenly). Place 1-inch apart on an ungreased baking sheet and bake until well risen and lightly browned, 20 to 22 minutes. Transfer to a wire rack to cool.
Meanwhile, whisk together the mayonnaise and chives in a small bowl. When the biscuits have cooled, cut each in half crosswise; spread with mayo and top with ham.