Place the butter in a heavy bottom sauce pot over medium heat. Melt until it foams up and swirl occasionally, cooking until you see brown bits appearing in the bottom. Swirl for another 20 seconds or so until more brown bits appear then pour into a stand mixer bowl. Be careful not to burn the butter. Allow the brown butter to cool for 30 minutes.
Whisk together the flour, baking soda and salt in a small bowl.
Add the sugars and vanilla to the stand mixer and beat with the paddle for 3-5 minutes to help cool the butter further. Add the egg and peanut butter and beat for another few minutes until it lightens up. If it still feels warm, allow to cool a bit more.
Add in the flour mix and mix on low just until combined well.
Heat the oven to 350 and line a few baking sheets with parchment paper. Use a 2 tbsp cookie scoop to make 6 balls per cookie sheet. Roll each scoop into a ball and make a well in the ball and add about 1/2 tsp jam and seal the ball. This is easier if you pull off a little bit of dough before making the well, flatten it and seal it over the well you made. Reshape into balls.
Bake each sheet for about 12 minutes, rotating half way through. Cookies should be just barely browned on the edges but you shouldn't see any raw dough on top. Cool for 10 minutes on the sheet then transfer to a rack to cool completely.