In a mixing bowl, whisk together the flour, baking powder, baking soda, and salt until fully combined. Set the dry mixture aside for now.
In a large mixing bowl, use a paddle attachment to cream the butter with the light brown and granulated sugars until smooth and well combined. This should take about 2-3 minutes on medium speed. Add the peanut butter and cream it into the mixture on medium speed for 1 minute until fully incorporated. Then, on low speed, mix in the egg and vanilla extract until smooth. Gradually stir in the dry ingredients, mixing until just combined.
Cover the cookie dough with plastic wrap and refrigerate for 30 to 60 minutes while the oven preheats. Chilling the dough is optional, but it helps reduce spreading during baking.
Preheat the oven to 350°F (177°C) and allow it to heat for 30 minutes before baking the cookies. Line a large baking sheet with parchment paper. Pour the granulated sugar into a small bowl and set aside.
Using a medium cookie scoop (about 2 tablespoons, 30 ml), scoop the peanut butter dough into balls. Roll each ball in the sugar, then place the dough balls onto the lined baking sheet, spacing them about 3 inches (8 cm) apart, as they will spread slightly.
Using your thumb or a teaspoon measuring spoon, press an indentation into the center of each cookie. Fill each well with a teaspoon (5 ml) of strawberry jelly or seedless jam.
Bake the cookies at 350°F (177°C) for 14-15 minutes or until the edges are set and cracks have formed across the tops of the cookies.
As soon as you remove the cookies from the oven, give them a “cookie scoot” to help reshape them into perfect rounds. To do this, place a biscuit cutter, small bowl, or large glass (slightly larger than the cookies) around each one and gently nudge the edges in a circular motion. This helps to refine the edges of the thumbprint, which may have spread during baking, and allows the jelly to firm up slightly. Let the cookies cool on the baking sheet for about 10 minutes.
If you like, spoon an additional teaspoon of jelly into each cookie while they are still hot.
To make these Peanut Butter and Jelly Cookies even more delicious, drizzle melted peanut butter over each cookie and sprinkle with chopped peanuts as they cool. Once the cookies have cooled to room temperature, transfer them to a cooling rack to cool completely and enjoy!