Combine shredded sharp Cheddar cheese with flour in a large bowl.
Cut butter with a sharp knife or pastry blender into 1/4-inch-thick slices. Sprinkle butter slices over flour mixture; toss butter with flour. Cut butter into flour with a pastry blender until crumbly and mixture resembles small peas. Cover and chill 10 minutes.
Combine buttermilk and diced pimiento; add buttermilk mixture to flour mixture, stirring just until dry ingredients are moistened.
Turn dough out onto a lightly floured surface; knead 3 or 4 times, gradually adding additional flour as needed. With floured hands, press or pat dough into a 3/4-inch-thick rectangle (about 9 x 5 inches). Sprinkle top of dough with additional flour. Fold dough over onto itself in 3 sections, starting with 1 short end. (Fold dough rectangle as if folding a letter-size piece of paper.) Repeat entire process 2 more times, beginning with pressing into a 3/4-inch-thick dough rectangle (about 9 x 5 inches).
Press or pat dough to 1/2-inch thickness on a lightly floured surface; cut with a 2-inch round cutter, and place, side by side, on a parchment paper-lined or lightly greased jelly-roll pan. (Dough rounds should touch.)
Bake at 450° for 13 to 15 minutes or until lightly browned. Remove from oven; brush with 2 Tbsp. melted butter.
Note: For testing purposes only, we used White Lily Self-Rising Soft Wheat Flour.
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