How to Make It
Heat your oven to 400. Prep a sheet pan with a sheet of parchment and have a second parchment sheet and pan ready.
Lay out your sheet of puff pastry and brush off any excess flour. Paint one half of the sheet with a thin layer of the beaten egg. Lay the anchovy filets in strips about one inch apart starting at the line where the egg starts and ending about a half-inch from the other side. Fold over the puff pastry and press firmly or roll with a rolling pin to adhere the two sides, being sure to press between the anchovy strips and really try and seal the open end.
Return the prepped dough to the fridge for 15-20 minutes, or up to a day.
Slice the puff across, so that each stick has little cross-sections of anchovy, about ¼ inch thick, and line up on your sheet pan with the cut side facing up so that you can see the little anchovy bits, about an inch apart. Top with the second sheet of parchment and the second sheet pan, which is going to help them stay thin and not puff too much and get super crispy.
Bake for 10 minutes, then remove the top sheet pan and top layer of parchment and check, you want to see that they are getting a little matte looking, and not shiny or greasy. Return to the oven, rotating the pan if one side seems to be cooking quicker, uncovered and keep baking for another 2-7 minutes until golden brown and crisp.
Cool pan completely on a rack and store in an airtight container for up to three days. If they lose their crisp, you can re-crisp in a 400-degree oven for 6-8 minutes.