Preheat your oven: Begin by preheating your oven to 375°F (190°C).
Cream the butter and sugars: In a large mixing bowl, combine 1/2 cup softened butter, 1/2 cup peanut butter, 1/2 cup sugar, and 1/2 cup brown sugar. Use an electric mixer or a wooden spoon to mix until the mixture is light and fluffy, about 2-3 minutes.
Add the egg and vanilla: Crack in 1 large egg and pour in 1/2 teaspoon vanilla extract. Mix until fully incorporated.
Combine the dry ingredients: In a separate bowl, whisk together 1 1/4 cups sifted all-purpose flour, 3/4 teaspoon baking soda, and 1/4 teaspoon salt. Gradually add this mixture to the wet ingredients, mixing until just combined. Do not overmix.
Roll the dough: Using your hands or a cookie scoop, roll the dough into balls, about 1 inch in diameter. Place them onto an ungreased cookie sheet, spacing them about 2 inches apart.
Flatten the cookies: Dip a fork in sugar (or flour) and press down gently on each ball to create a crisscross pattern on top, which also helps flatten the cookies.
Bake: Place the cookie sheet in the preheated oven and bake for 10-12 minutes, or until the edges are lightly golden brown. The centers may look slightly underbaked, but they will firm up as they cool.
Cool and enjoy: Remove from the oven and let the cookies cool on the sheet for a few minutes before transferring them to a wire rack to cool completely. Enjoy your delicious peanut butter cookies!