Whisk the all-purpose flour, whole-wheat flour, baking powder and salt in a medium bowl. Beat the butter, granulated sugar, brown sugar and vanilla in a large bowl with a mixer on medium-high speed until light and fluffy, about 5 minutes. Beat in the egg until combined, then beat in the peanut butter until smooth. Reduce the mixer speed to low and beat in the flour mixture until just combined. Turn out the dough onto a piece of parchment paper and pat into a rectangle. Top with another piece of parchment and roll out the dough into a 1/4-inch-thick rectangle. Refrigerate until firm, about 1 hour.
Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Trim the edges of the dough and cut into 1 1/2-inch squares. Arrange 2 inches apart on 2 unlined baking sheets; press each with the back of a fork to make a crisscross pattern, flattening slightly. Gather the scraps and refrigerate until firm, if needed; reroll to cut out more cookies.
Bake, switching the pans halfway through, until the cookies are almost set, about 14 minutes. Let cool 5 minutes on the pans, then transfer to racks to cool completely. Sandwich the grape jelly (about 1 teaspoon per sandwich) between the cookies.