Preheat oven 375. Place cornstarch, lemon juice, water and sugar in saucepan and whisk until cornstarch is dissolved. Add cherries, turn heat to med high and stirring, bring to boil. Reduce heat to med low and simmer until cherries have softened and mixture has thickened, 10-12 min. Remove from heat and let cool completely.
Place 1 sheet of phyllo on dry work surface and brush lightly with some of the oil. Layer another sheet on top. Cut phyllo stack in half length wise. Spoon about 1/4 C of filling onto one of the phyllo stacks about 2 inches from top. Fold the top corner of the phyllo stack over the filling to form a triangle shaped pocket. Continue to fold the pocket down the strip of phyllo in a way that maintains the triangle shape. Repeat with another 1/4 C of filling and the second stack of phyllo, then brush the outside of each triangle lightly with oil. Repeat with the remaining phyllo, filling and oil.
Place turnovers on baking sheet. Bake until crisped and browned, about 20 min.l
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