Line a 6-qt slow cooker with nonstick foil; if using regular foil, butter the foil to prevent sticking.
In a large bowl, whisk together the melted unsalted butter and 1/2 cup chunky peanut butter until smooth.
Add 1 1/2 cups granulated sugar, 2/3 cup unsweetened cocoa powder, the lightly beaten eggs, and 2 tsp vanilla extract; whisk until combined.
Stir in 1/3 cup all-purpose flour and 1/2 tsp salt until just combined; do not overmix.
Fold in 2/3 cup semi-sweet chocolate chips and 2/3 cup peanut butter chips.
Pour the batter into the prepared slow cooker and spread it into an even layer.
Cover the slow cooker with the lid and place a paper towel between the lid and insert to absorb condensation so steam doesn't drip onto the brownies.
Cook on low for 2 1/2 to 3 hours (about 150–180 minutes) until the edges are set and a toothpick inserted near the center comes out with a few moist crumbs.
While the brownies cook, make the peanut butter sauce: combine 3/4 cup creamy peanut butter, 3/4 cup heavy whipping cream, 1/3 cup granulated sugar, and 1/4 cup light corn syrup in a medium saucepan (reserve the 1/2 tsp vanilla).
Cook the sauce over medium heat, stirring frequently, until the sugar has dissolved and the mixture is smooth.
Remove the sauce from heat, stir in 1/2 tsp vanilla extract, and let it cool slightly before serving or drizzling over brownies.
Serve the brownies hot or cool to room temperature before slicing.