Preheat the oven to 375°F. Line a large baking sheet with parchment paper or use a silicone baking mat. Set aside.
In a mixing bowl, whisk together the flour, baking powder, baking soda, and salt.
In another bowl, using an electric mixer at medium speed, beat together peanut butter, softened butter, light brown sugar, and golden caster sugar until smooth and creamy. Scrape the bottom and the sides of the bowl to make sure all ingredients are well combined. Beat in the egg and vanilla extract for an additional 1-2 minutes until light and fluffy. Then, add dry ingredients and finely chopped peanuts. Beat on low until combined. Cover the bowl with a plastic wrap and let chill in the refrigerator for 1 hour.
Remove the cookie dough from the refrigerator. Scoop the dough and roll it out into 1-inch balls. Place on the prepared baking sheet or silicone baking mat leaving about 1-2 inches between each cookie. The cookies will spread during the baking process.
Make a well in the center of each cookie using the back of a measuring teaspoon. Then, spoon raspberry jam into each well. Bake in the preheated oven for 10-12 minutes or until the edges are golden and set. Remove from the oven and let cool for 5-10 minutes on the baking sheet. Then, transfer to a cooling rack. Optionally, sprinkle with additional finely chopped peanuts or drizzle with peanut butter. Enjoy!