Heat the oven to 350 degrees and line your baking sheets with parchment paper to aid in the clean up.
In a large bowl with an electric mixer on medium speed, beat together the peanut butter and the softened butter until they are smooth and creamy.
Add the brown sugar and the white sugar and beat them together at high speed until they are light and fluffy. This should take about two minutes.
Add both eggs and the vanilla, beating them into the mixture on medium speed until they are blended.
Add the flour, baking soda, and salt, beating the ingredients together until the dough starts to stick together.
Scoop the dough out with a tablespoon into walnut-sized blobs. Using clean, lightly floured hands, roll the dough into balls, placing them on the baking sheet two or three inches apart.
Dip a fork into a bit of flour and lightly press the tops of the cookies down with the tines of the fork creating the distinctive criss-cross pattern on the surface of the cookies.
Bake the cookies for 10 to 12 minutes until the edges are just set and slightly brown. The centers may be a little glossy and tacky, but leave the cookies on the baking sheet for about five minutes, and the centers will set up a bit more.
Gently scrape the cookies off the baking sheet to a wire rack to cool for another fifteen minutes. Store the cookies in an airtight container for up to a week.
These cookies also freeze very well. To freeze the cookies, allow them to completely cool by setting them on a plate for about an hour. Place the cookies gently in stacks of two or three and put the stacks in a zip-top freezer bag. Press out all of the air and freeze the cookies for up to three months. To thaw, place the cookies in the refrigerator overnight before eating.