Preheat oven to 350°F. Line baking sheets with parchment paper or silicone mats.
Whisk together flour, baking soda, baking powder, and salt in a medium bowl until fully combined.
Beat softened butter, peanut butter, brown sugar, and granulated sugar on medium speed until light and fluffy, about 2-3 minutes. Scrape down the sides as needed.
Add the egg and vanilla extract and beat on medium speed until fully incorporated, about 1 minute. Scrape the bowl again.
Add the flour mixture in two additions, alternating with the milk, mixing on low speed after each addition just until combined. Do not overmix.
Using a medium cookie scoop, drop rounded portions of batter onto the prepared baking sheets, spaced 2 inches apart. Gently smooth the top of each mound.
Bake one sheet at a time on the center rack for 10-12 minutes until the cakes spring back when touched. Cool on the sheet for 5 minutes, then transfer to a wire rack to cool completely.
In a clean bowl, beat softened butter and peanut butter until smooth. Gradually add powdered sugar, then add heavy cream and a pinch of salt. Beat on medium-high for 2 minutes until light and fluffy.
Once cakes are completely cool, flip half of them flat side up. Pipe or spread a generous layer of peanut butter filling around the edge, leaving a small well in the center. Spoon about 1 teaspoon of jelly into the center well.
Top each filled cake with a second cake, flat side down, and gently press until the filling reaches the edges. Serve right away or store for later.