Position a rack in the center of the oven and preheat to 350 degrees F. Butter the bottom and sides of a 10-inch cast iron skillet with 1 tablespoon of the butter. Beat the remaining butter and sugar together in a large bowl with an electric mixer on medium speed until light and fluffy, scraping down the bowl as needed. Add the egg, peanut butter and vanilla and beat until smooth and creamy. Add the salt and baking soda and beat until combined. Reduce speed to low and gradually add the flour, mixing until the dough comes together and is smooth.
Use a rubber spatula to scrape the dough into the prepared skillet, gently spreading it to the edges and smoothing the top. Make a crosshatch pattern on the surface of the dough with the corner tip of the rubber spatula. Sprinkle the top with sugar.
Bake until the edges are brown and crisp and the center is still soft, 40 to 45 minutes. Cool on a wire rack for 30 minutes (the cookie will firm as it cools). Warm the jam in the microwave, about 45 seconds. Slice cookie into wedges and serve with warm jam.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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