Preheat the oven to 350°F (175°C). Line an 8×8 inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.
In a large mixing bowl, combine the peanut butter and granulated sugar. Stir until smooth and creamy.
Add the egg and vanilla extract to the peanut butter mixture and mix until fully incorporated.
Sprinkle the baking soda and salt over the mixture and stir until just combined.
Spread the batter evenly into the prepared baking pan, smoothing the top with a spatula.
Bake in the preheated oven for 18-20 minutes, or until the edges are lightly golden and the center is set.
Remove from the oven and allow the bars to cool completely in the pan on a wire rack.
Once cooled, use the parchment paper overhang to lift the bars out of the pan and cut into 16 squares.