Preheat your oven to 350°F (175°C) and line two large baking sheets with parchment paper.
In a large bowl, cream together the peanut butter, softened butter, granulated sugar, and brown sugar until the mixture is light and fluffy, about 3 minutes.
Beat in the egg and vanilla extract until fully incorporated, scraping down the sides of the bowl as needed.
In a separate small bowl, whisk together the flour, baking powder, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined.
Scoop rounded tablespoons of dough and roll them into 1-inch balls, placing them 2 inches apart on the prepared baking sheets.
Use your thumb or the back of a rounded teaspoon to press a deep indentation into the center of each ball.
Bake for 10 to 12 minutes, or until the edges are lightly golden and the centers look set.
Immediately after removing the tray from the oven, gently press the centers again with your spoon if they have puffed up during baking.
In a small saucepan over low heat, melt the caramel bits with the heavy cream, stirring constantly until smooth.
Spoon approximately 1 teaspoon of the melted caramel into the center of each warm cookie.
Allow the cookies to cool completely on the baking sheet to let the caramel set before serving.