Preheat oven to 350°F. Line a baking sheet with parchment paper.
In a large bowl, combine the softened butter and sugar. Mix until just combined and smooth—do not overcream.
Add the eggs, egg yolk, and vanilla bean paste. Mix until fully incorporated.
In a medium bowl, whisk together the flour, baking powder, and salt.
Gradually add the dry ingredients and mix just until a stiff dough forms.
Divide dough in half. Shape each portion into a log about 10–11 inches long and 2-2½ inches wide. Place logs several inches apart on the prepared baking sheet.
First bake: Bake for 25–30 minutes, until the logs are set and lightly golden. Remove from oven and let cool for 10–15 minutes.
Reduce oven temperature to 325°F.
Using a serrated knife, slice the logs diagonally into ¾-inch thick pieces.
Arrange slices cut-side down on the baking sheet.
Second bake: Bake for 12–15 minutes, flip each biscotti, then bake an additional 12–15 minutes until dry and crisp.
Transfer biscotti to a wire rack to cool completely. They will continue to crisp as they cool.