Preheat oven to 450ºF.

Pulse 2 cups of flour with the cheese two or three times in a food processor fitted with the knife or dough blade. Set aside remaining 1/4 cup of flour. Scatter the 1/4-inch shortening pieces over the flour mixture and pulse 2 or 3 times. Scatter the 1/2-inch shortening pieces over the flour mixture and pulse 2 or 3 times until mixture resembles well-crumbled feta cheese, with no piece larger than a pea. Add the pimentos, onion, and 3/4 cup of buttermilk. Set aside the remaining 1/4 cup buttermilk. Pulse mixture briefly to incorporate into a shaggy wettish dough. When the blade stops, remove the lid and feel the dough. Add reserved buttermilk or flour as needed to make a slightly wettish dough. Pulse once or twice more until the dough looks shaggy but holds together.

Lightly sprinkle a board or other clean surface with some of the reserved flour. Turn the dough out onto the board and sprinkle the top lightly with flour. With floured hands, fold the dough in half, and pat dough out into a 1/3- to 1/2-inch-thick round, using a little additional flour only if needed. Flour again if necessary, and fold the dough in half a second time. If the dough is still clumpy, pat and fold a third time. Pat dough out into a 1/2-inch-thick round for a normal biscuit, 3/4-inch for a tall biscuit, and 1-inch-thick for a giant biscuit.

Brush off any visible flour from the top. For each biscuit, dip a 2-inch biscuit cutter into the reserved flour and cut out the biscuits, starting at the outside edge and cutting very close together, being careful not to twist the cutter. The scraps may be combined to make additional biscuits, although these scraps make rougher biscuits.

Using a metal spatula if necessary, move the biscuits to a lightly greased baking sheet so they are touching each other. Bake the biscuits on the top rack of the oven for a total of 13 to 15 minutes until light golden brown.

After 6 minutes, rotate the pan in the oven so that the front of the pan is now turned to the back, and check to see if the bottoms are browning too quickly. If so, slide another baking pan underneath to add insulation and retard browning. Continue baking another 7 to 9 minutes until the biscuits are light golden brown. When the biscuits are done, remove from the oven and lightly brush the tops with softened or melted butter. Turn the biscuits out upside down on a plate to cool slightly.

Serve hot, right side up.