Preheat the oven to 350 degrees. Line baking sheets with parchment paper.
In a medium bowl, beat the peanut butter, sugar, vanilla and egg until well combined.
Scoop the dough by the tablespoonful onto prepared baking sheets. Flatten the mounds with the tines of a fork, making a crisscross pattern on the cookies. Sprinkle coarse salt on top of the cookies.
Bake until golden around the edges, about 8-10 minutes, switching the position of the baking sheet halfway through baking. Remove the pan from the oven to a rack, cool cookies right on the pan. Repeat with the remaining dough. Store cooled cookies in an airtight container.
Cook's Note: Stir in 1/2 cup chocolate chips OR 1/2 cup chopped peanuts if desired.