Preheat the oven to 350°F (175°C). Grease a 24-cup mini muffin pan well with nonstick spray or a little oil.
In a large bowl, whisk together the all-purpose flour, baking powder, and salt until evenly combined.
In a medium bowl, whisk the peanut butter, egg, and granulated sugar until smooth. Stir in the milk and vanilla until combined.
Pour the wet mixture into the dry ingredients and stir with a whisk just until the batter is smooth; do not overmix.
Spoon the batter into the prepared mini muffin cups, filling each about three-quarters full.
Bake for 8–10 minutes, until the tops spring back when lightly pressed and a toothpick inserted into the center comes out clean.
Let muffins cool in the pan on a rack for about 5 minutes, then use a butter knife to gently loosen and transfer them to the rack to cool completely, about 20 minutes.
Slice each muffin in half horizontally, spread about 1/2 teaspoon jam on the inside of each bottom half, and replace the top to make mini sandwich muffins.