Combine dry ingredients in a bowl and using a pastry cutter, cut in butter until mixture resembles coarse crumbs. Add cheese and stir to combine.
Combine eggs and heavy cream. Add the egg mixture to the flour mixture and stir just until it forms a dough.
On a lightly floured surface, knead dough by folding and gently pressing dough for 10 to 12 strokes until nearly smooth. Pat dough into an 8-inch circle and cut into 8 wedges.
Place wedges 1-inch apart on an ungreased baking sheet. Brush with milk.
Bake for 18-22 minutes or until golden. Remove scones from the baking sheet. Let cool slightly.
Combine chive butter ingredients. Brush over warm scones and serve.