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Preheat oven to 350°F.
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Beat butter and sugar in a large bowl until light and fluffy.
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Add egg and vanilla and beat to combine.
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Whisk together flour, baking powder, and salt. Add to creamed mixture and beat until just combined. Dough will be slightly crumbly.
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Divide dough into 4 equal portions and press each together. On a lightly floured surface, roll each portion of dough to a log 12 inches long.
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Place logs on 2 ungreased cookie sheets (2 logs per sheet about 3 inches apart). Use your finger to to make an indention in the top of each log about 1/2-inch wide and 1/4-inch deep.
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Place dulce de leche in a plastic zip top bag. Cut a corner from the bag. Squeeze dulce de leche from bag to pipe 2 tablespoons into the indention on each log.
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Bake for 18 to 20 minutes, or until light golden.
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Cool logs on cookie sheets for 2 minutes. Use 2 spatulas to carefully lift logs to a cutting board.
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Slice logs diagonally into 3/4-inch wide strips. You may need to scrape dulce de leche off sides of the knife as you are cutting.
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Place cookie strips on a wire rack to cool completely.