Preheat an oven to 350 degrees F (175 degrees C); line a baking sheet with parchment paper.
Beat the butter and sugar with an electric mixer in a large bowl until smooth; add the vanilla and continue to beat.
Mix in the flour mixture until just incorporated.
Roll dough into 1 1/2-inch balls and arrange on prepared baking sheets. Make a small hole in the center of each ball, using your thumb and finger; fill the holes with grape jam.
Bake in the preheated oven until lightly browned, about 14 minutes; allow to cool on baking sheet for 1 minute.
Put the peanut butter in a microwave-safe bowl, and cook in a microwave oven until soft for 10 - 30 seconds, checking every 10 seconds. Be careful not to overheat the peanut butter; do not let it bubble. Stir the vegetable oil through the heated peanut butter; drizzle over the warm cookies.
