In a food processor combine all ingredients except the olive oil. Scrape down the sides as necessary. With the machine running, slowly poor in the olive oil. Add more or less than listed according to desired consistency. Add more salt if needed.
Will keep in the fridge for one week.
To make the pesto straws:
Preheat oven to 400°F. Line a sheet pan with parchment paper.
On a floured surface, roll out one sheet of the puff pastry to about 2 inches larger on each side. Cut lengthwise into two rectangles. Spread about ⅓ to ½ cup of the pesto onto one half of the puff pastry. You want it to be covered, but don't use to much or the straws won't stay together. Put the other half of the puff pastry on top and pat firmly with your hand to make sure they stick together. Cut into ¾ inch long strips. Pinch the edges of each strip together.
Pick up each strip and twist it a few times, then place it on the prepared sheet pan. Very lightly brush each twist with the whisked egg and sprinkle with some parmesan cheese.
Bake for about 15 minutes or until golden brown. Cool on wire rack and serve immediately.