In a large bowl, mix together flour, baking powder, and salt. Using a pastry blender, cut in shortening and butter until mixture resembles coarse crumbs. Add cheese and pepper; mix well. Make a well in center of the mixture and add milk all at once; stir until just moistened.
Turn the dough out onto a lightly floured surface. Knead by folding and gently pressing dough for 10 to 12 strokes or until almost smooth. Divide dough in half. Roll or pat each half into a 6-inch square. Using a sharp knife, cut dough into 2-inch squares.
In a small bowl, whisk together egg and the water; brush onto tops of biscuits. Place on a large greased baking sheet.
Bake in a preheated 400-degree oven for 13 to 15 minutes or until golden. Transfer to a wire rack. Serve while warm.
NOTE: To make these ahead, prepare as directed and cool completely. Place biscuits in a freezer container or freezer bag and seal, label, and freeze for up to 3 months. When ready to serve, wrap biscuits in foil and bake for 20 to 25 minutes in a 300-degree oven.
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