Preheat the oven to 350 degrees F and place the racks in the upper and lower third of the oven.
In a medium bowl, mix the peanut butter, sugar, vanilla and egg until well combined. Spoon 1 tablespoon of the mixture about 1 inch apart onto ungreased baking sheets. Flatten the mounds with the tines of a fork, making a crosshatch pattern on the cookies. Sprinkle coarse salt on top of the cookies.
Bake until golden around the edges, about 10 minutes, switching the position of the sheets halfway through baking. Transfer to racks to cool. Repeat with the remaining dough.