Preheat the oven to 350ºF. Line an 8x8 inch baking dish with parchment or foil.
In a medium bowl, stir together the flour, oats, and baking soda.
Using a mixer, cream together the butter and sugar until soft and smooth. Add the egg, peanut butter, and vanilla. Mix until smooth again.
Pour the flour and oat mixture into the bowl with the peanut butter mixture. Use a spoon to stir until everything is evenly combined and a thick cookie dough forms.
Press 3/4 of the cookie dough into the bottom of the prepared baking dish in an even layer. Make sure there are no holes or gaps and the dough is compact. Spread the jam evenly over the surface of the cookie dough. Drop the remaining cookie dough over top of the jam in small pieces or crumbles. Sprinkle the peanuts over the top and press them down gently into the jam and dough crumbles. The crumbles and peanuts will not cover the jam in a solid layer.
Cook the peanut butter and jelly bars for 35-40 minutes, or until the top is golden brown. Let the bars cool slightly, then use the parchment to lift the bars out of the dish. Cut the bars into 12 pieces and enjoy.