Set your oven to 350°F and prepare two baking sheets by lining them with parchment paper or silicone mats.
In a large mixing bowl or the bowl of a stand mixer, beat 1/2 cup of softened butter until smooth and creamy. Make sure to scrape down the sides of the bowl.
Add 3/4 cup of granulated sugar and 1/4 cup of packed brown sugar. Mix thoroughly until the mixture is fluffy, ensuring there are no butter lumps by scraping the bowl as needed.
Incorporate 1 cup of peanut butter into the mixture (smooth or crunchy works, but smooth is recommended). Blend well until fully combined.
Mix in 1 large egg and 1 teaspoon of vanilla extract, ensuring everything is well combined.
Add 1 1/2 cups of all-purpose flour to the bowl but don't mix yet. Sprinkle 1 teaspoon of baking soda and 1/4 teaspoon of kosher salt over the flour. Lightly stir these dry ingredients together before turning on the mixer.
Use the mixer to combine the dry and wet ingredients until just incorporated. Scrape the bowl's sides and bottom to ensure all the flour is mixed in, then stop once you no longer see streaks of flour. Avoid overmixing to keep the cookies soft.
Roll the dough into 1 1/2-inch balls. Place the dough balls on the prepared baking sheets, spacing them a couple of inches apart. Each 11x17-inch pan should fit about 12 cookies.
Dip a fork in sugar or flour to prevent sticking, then gently press it into each dough ball to create a crisscross pattern. Flatten the cookies to about 1/2-inch thick, re-dipping the fork as needed.
Stir 1/2 cup of seedless jam to remove lumps, then use a teaspoon to add a small dollop (about 1 teaspoon or less) to the center of each cookie. Be careful not to let the jam spill over the edges, as it may spread during baking.
Bake the cookies for 12-15 minutes, or until the edges are set and slightly golden. Avoid underbaking, as these cookies are delicate due to the jam.
Allow the cookies to cool on the baking sheet for 5-10 minutes before transferring them to a wire rack to cool completely.
For an extra touch, you can add more stirred jam to the tops of the cooled cookies for presentation. Drizzle with melted peanut butter and sprinkle with chopped peanuts if desired.