Step 1: In a large bowl, cream together the softened unsalted butter and granulated sugar using an electric mixer until light and fluffy, about 2-3 minutes.
Step 2: Gradually drizzle in the extra virgin olive oil while continuing to mix. Beat until fully combined and the mixture is smooth.
Step 3: Add the finely chopped fresh rosemary, vanilla extract, and salt to the butter and olive oil mixture. Mix until just combined.
Step 4: In a separate bowl, whisk together the all-purpose flour and baking powder. Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined and a soft dough forms. Be careful not to overmix.
Step 5: Turn the dough out onto a lightly floured surface. Gently knead it a few times to bring it together into a cohesive ball.
Step 6: Divide the dough in half. Shape each half into a disc about 1/2 inch thick. Wrap each disc tightly in plastic wrap and refrigerate for at least 30 minutes, or up to 2 days.
Step 7: Preheat your oven to 325°F (160°C). Line a baking sheet with parchment paper.
Step 8: On a lightly floured surface, roll out one chilled dough disc to about 1/4 inch thickness. Cut into desired shapes (squares, wedges, or rounds) using a sharp knife or cookie cutters. Prick the tops of the shortbread with a fork (docking) to prevent puffing.
Step 9: Arrange the cut shortbread pieces on the prepared baking sheet, leaving about 1 inch between them. Repeat with the second disc of dough.
Step 10: Bake for 18-25 minutes, or until the edges are lightly golden and the center appears set. Shortbread should remain pale. Baking time may vary with size and thickness.
Step 11: Remove from the oven and, if desired, sprinkle lightly with a pinch of flaky sea salt immediately while still warm. Let cool on the baking sheet for 5-10 minutes before transferring to a wire rack to cool completely.