Arrange racks in upper and lower thirds of oven; preheat to 350°F. Line 2 rimmed baking sheets with parchment paper.

Beat egg with a whisk in a medium bowl. Add peanut butter and sugar and whisk until fully incorporated and smooth. Drop mounds of dough by the tablespoonful onto parchment-lined baking sheets, spacing about 2" apart. Lightly press tops of cookies with tines of a fork, making a crosshatch pattern. Sprinkle with salt, if using. Bake cookies, rotating trays halfway through, until golden brown, 10–12 minutes. Transfer to a wire rack and let cool slightly.

Cookies can be baked 3 days ahead. Store in an airtight container at room temperature.