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Heat the oven to 350°F. Line the bottom of a 9- x 13- inch baking pan or dish with parchment paper so that there are “handles” overhanging on both short ends. Use an electric mixer to beat together the oil, milk, and sugars until light and fluffy, then beat in the peanut butter. Add the vanilla.
Stir together the flour, baking soda, and salt in a small bowl. Gradually add the dry ingredients to the peanut butter mixture and stir with a rubber spatula until just combined.
Press the dough evenly into the prepared pan or dish. (Your hands are the ideal tool for this.) Bake until puffed, lightly browned around the edges, and firm to the touch, 30 to 35 minutes.
Transfer the pan to a wire rack and immediately spread the jam evenly on top. Let the cookies and jam cool completely in the pan. To slice, move the pan near a clean work surface, grab both overhanging ends of the paper, and move everything to the work surface. If you’d like, slip the paper out from underneath before cutting the cookies into pieces of any shape. Eat right away or refrigerate in an airtight container for several days (they also freeze well for a couple months).