Brown the butter: In a medium saucepan over medium heat, melt the butter, swirling the pan occasionally, until golden brown flecks form on the bottom and it smells nutty, about 5 to 7 minutes. Pour into a bowl and let it cool until just slightly warm.

Mix the wet ingredients: Whisk the browned butter and brown sugar together until smooth, then beat in the egg and vanilla extract.

Combine the dry ingredients: In a separate bowl, whisk together the graham cracker crumbs, flour, baking soda, and salt. Stir the dry ingredients into the wet ingredients until just combined.

Chill the dough: Cover and refrigerate the dough for 20 to 30 minutes so the cookies hold their shape while baking.

Bake the cookies: Preheat the oven to 350 degrees F. Scoop the dough into 3 tablespoon portions, roll into balls, and place on parchment lined baking sheets, flattening slightly. Bake for 10 to 12 minutes, until the edges are set and the centers still look soft. Let cool completely.

Make the frosting: Beat the cream cheese and butter together until creamy. Add the powdered sugar, vanilla extract, and a pinch of salt, then beat until smooth and fluffy.

Assemble the cookies: Frost each cooled cookie generously with the cream cheese frosting. Warm the jam slightly to make it easier to spread, then spoon a small swirl onto the center of each cookie.