Line an 8x8 baking dish with parchment paper. Set it aside.

Melt the chocolate chips and peanut butter in a double boiler. Stir until the mixture is smooth. Set it aside.

In a large pot, melt the butter over medium heat. Stir in the sugar until the mixture is smooth. Stir in the salt and vanilla extract. Turn the heat off and stir in the oats.

Transfer two-thirds of the oat mixture into the baking dish and press it down firmly into an even layer to create a solid crust. Pour three-quarters of the chocolate mixture onto the oat crust and spread it into an even layer. Crumble the remaining oat mixture on top of the chocolate mixture in an even layer. Drizzle the remaining chocolate mixture over the top. Transfer the bars to the refrigerator to set, about 2 hours.

Cut the bars into squares and serve immediately. Store in the refrigerator for up to 5 days.