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Line large baking sheet with parchment paper.
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In large bowl, sieve or sift the flours, sugar, baking powder, salt, garlic and onion powders, and cayenne or paprika together. Drop the cubes of butter into the dry ingredients and toss to coat the butter. Cut in the butter with a wire pastry cutter until mixture resembles coarse crumbs or pea-sized pieces. Quickly run fingers through the mixture several times, scooping up bits of the butter here and there and rubbing them into long paper-thin slivers between the thumb and forefinger. It is not necessary to do this with every single piece of butter – just quickly pick several at random.
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Stir in the shredded cheese, diced ham, and chopped chives. Make a well in the center of the dry ingredients.
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In a small bowl or large measuring cup, lightly whisk the egg, whipping cream, and milk (or 10%MF cream) together, just until blended. Pour liquid ingredients all at once into well in center of dry ingredients. Using a fork, stir dough just enough that the liquid is absorbed into the dry ingredients and the dough can be roughly brought together. Dough will be soft and sticky and some floury spots may remain. Transfer dough onto lightly floured work surface, gently working it just until the dry ingredients are barely incorporated and a shaggy dough mass forms. Bring dough together into a ball.
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Roll, or pat, the dough into a rectangle about ¾” (apx. 1.9cm) thick. Brush lightly with half of the melted butter. With dough positioned so the short ends of the rectangle are facing you, and with the aid of a large bench scraper, fold one third of the dough up from the bottom short edge into the center and then fold the other top third down onto the top of the folded two-thirds of dough, just as if folding a business letter. Roll, or pat, the dough down into another rectangle, again about ¾“ (apx. 1.9cm) thick. Brush with remaining melted butter. Fold the dough in half, this time from one of the long sides, finally rolling, or patting, the dough into a neat, tidy rectangle about ¾“ – 1” (apx. 1.9cm – 2.54cm) in thickness.
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Using a sharp knife or bench scraper, cut the dough into apx. 14-16 equal-sized squares, apx. 2” wide, depending on actual thickness of dough. Place scones, about 1” (2.54cm) or so apart, on prepared baking sheet and place in freezer for about 20 minutes (or in refrigerator for 30-35 minutes) to chill and allow butter to firm up.
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To make square- or round-shaped scones using a sharp-edged cookie cutter:
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Cookie cutters, instead of a knife or bench scraper can be used to cut out scones. Use a lightly floured, sharp-edged, 2” (5.08cm) square or round cutter to cut scones, as close together as possible, from dough. Push the cutter straight down and out of the dough without twisting it in the process. Re-flour the cutter before cutting out each scone. Gather dough scraps and form into a rectangle from which to cut remaining scones, being careful to work the dough no more than absolutely necessary to bring it together. Transfer scones to prepared baking sheet, placing scones about 1” (2.54cm) apart. Refrigerate for 30-35 minutes (or place in freezer for about 20 minutes) to firm up.
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For either square or round-shaped scones, remove scones from refrigerator/freezer and, using a pastry brush, lightly brush the reserved whipping cream on tops of scones, being careful not to allow any cream to drip down over the sides of the scones. Alternatively, melted butter may be used to brush scone tops, if desired.
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To bake scones (square- or round-shaped):
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While scones are chilling, position oven rack in center of oven and preheat oven to 400°F. Bake scones apx. 16-18 minutes, or until they are just light golden brown on the edges and tops are golden tanned. Rotate the baking sheet halfway through the baking time. Remove scones from oven and leave them on the baking sheet for 3-4 minutes before transferring them to a wire rack.