Preheat the oven to 350°F (175°C). Line large rimmed baking sheets with parchment paper.

In a large microwave-safe bowl, microwave the butter in 20-second bursts until melted. Stir in the peanut butter until well combined.

Add in both sugars and whisk together. Set aside to cool until just warm.

In a medium bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.

To the butter mixture, add the eggs, milk, and vanilla. Whisk very well until thoroughly combined.

Add the flour mixture to the wet mixture and stir with a sturdy spatula until combined.

Using a large spring-loaded cookie scoop, form the dough into 3-tablespoon rounds. Roll between your palms to smooth.

Using your thumb, create a well in the center of each dough ball. Add 1 heaping teaspoon of chilled preserves and partially seal it with dough, keeping some of the jam peeking out.

Place on the baking sheet with the visible jam facing up, spacing about 3 inches apart.

Bake for about 14 minutes, or until puffed up and lightly brown at the edges.

Let cookies cool for 5 minutes on the baking sheet before transferring to a cooling rack to cool completely.