Preheat oven to 350 degrees F. In the bowl of a food processor pulse flour, salt and baking powder to combine. Add the shortening and pulse until the mixture resembles coarse meal. With machine running, gradually add between 1/2 and 3/4 cup ice water until the dough comes together, about 1 minute.
Transfer dough to a lightly floured surface. Form into a smooth rectangle, about 4 by 6 inches, with your fingers. Then roll out the dough into an 8 by 10-inch sheet, 1/4-inch thick. Cut dough lengthwise into 1/4-inchwide strips.
Using your hands, gently roll each strip back and forth into 16-inch long sticks. For the twisted version, grab each end of the dough strip with your fingers and carefully stretch and twist the strip in opposite directions. Arrange sticks on 2 baking sheets, side by side but not touching. Lightly brush with olive oil. Press ends onto the baking sheet to keep the sticks straight while they bake.
Bake until firm and cooked through, 14 to 18 minutes. Transfer to a wire rack to cool. Wrap a slice of prosciutto around each breadstick. Arrange on a platter and serve.
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