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1. For pastry, combine flour, sugar and 1 teaspoon salt in food processor; pulse to blend. Add butter and shortening; pulse until butter is the size of small peas. Combine 3 tablespoons water and the vinegar in a cup. Add liquid through feed tube, pulsing just until dough starts to come together (add remaining 1 tablespoon water if necessary). Gather dough into a ball; press into a disk. Wrap well in plastic wrap and refrigerate 30 minutes.
2. Meanwhile, for filling, heat oil in large skillet over medium heat. Add onion and garlic and cook just until softened, 3 minutes. Increase heat to medium-high; add artichokes, tomatoes, thyme, 1/2 teaspoon salt and pepper; cook 2 to 5 minutes more, until tomatoes are softened and mixture is slightly golden. Cool. Transfer filling to processor and pulse until very finely chopped. Transfer to bowl and stir in cheeses.
3. Heat oven to 375 degrees F. Lightly grease 2 large jelly-roll pans or cookie sheets.
4. Unwrap pastry. On a lightly floured surface, roll pastry into a 14-1/2-inch circle, 1/8-inch thick. With a 3-inch round biscuit cutter or top of a glass, cut out as many circles as possible, rerolling scraps. Reroll each circle until 3-1/2 inches in diameter. Fill center of each circle with a slightly mounded 1 tablespoon of filling; fold circles in half and press along edges. Brush tops with egg; transfer to prepared pans.
5. Bake empanadas 22 to 28 minutes, switching pans once halfway through, until golden.