Add the butter, cooked potatoes, caramelized onion, fresh dill, and chives. Use a pastry cutter or fork to break up butter into small, pea-sized pieces. Try to work quickly so as not to warm up the butter.
Turn the dough onto a lightly floured surface and dust the top of the dough lightly with additional flour. Fold the dough over and onto itself 3 times to create layers of dough (this is what creates flakiness).
Form into a 1-inch-thick disc. Use a biscuit cutter (or similarly sized rim of a drinking glass) to cut straight down through the dough. Try not to twist cutter, as this will seal off the dough and prevent flakiness from being created.
Repeat and place the biscuits on a baking sheet, making sure they are touching (this helps them rise uniformly).
Gently re-form the dough scraps. Cut out 1 or 2 more biscuits and repeat until all of the dough is used.