Preheat the oven to 350°F and line two rimmed baking sheets with parchment paper.
In a large microwave-safe bowl, combine the peanut butter and butter. Microwave on HIGH in 30-second intervals until the butter is almost melted, about 1 minute. Whisk until you have a smooth, cohesive peanut butter.
Add the brown sugar, egg, vanilla extract, baking soda, and salt and whisk until smooth. Using a spatula, fold in the flour until just combined. If possible, let the dough chill in the refrigerator for at least 30 minutes and up to overnight.
Scoop level tablespoons of the dough and form into balls (see Note). Roll each ball in crushed peanuts or demerara sugar, if desired, and transfer to the prepared baking sheets, spacing about 2 inches apart.
Using a ½ teaspoon measuring spoon, create a small indentation in the center of each cookie. Bake the cookies until puffed and slightly browned around the edges, about 10 minutes. Remove from the oven and transfer the baking sheets to a cooling rack. Once again, use the ½ teaspoon to press an indentation into each cookie.
In the jam jar or a microwave-safe measuring cup, microwave the jam on HIGH in 30-second intervals until it is runny, about 1 minute. Spoon ½ teaspoon of jam into the indentation of each cookie and then let cool completely before serving. Enjoy!