Preheat your oven to 350°F (175°C). Line a 9×13-inch baking pan with parchment paper, leaving some overhang for easy removal.
In a large bowl, cream together the softened butter, peanut butter, granulated sugar, and brown sugar until light and fluffy, about 3-4 minutes.
Beat in the eggs one at a time, then stir in the vanilla extract. Mix until just combined.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add this dry mixture to the wet ingredients, mixing on low speed until just combined.
Press the dough evenly into the prepared baking pan. For the authentic texture, don't compress the dough too firmly – a light touch with slightly damp hands works perfectly.
Bake for 18-20 minutes until the edges are lightly golden and the center appears set but still soft.
While the bars are cooling, make the frosting. In a medium saucepan over low heat, melt the butter and peanut butter together until smooth.
Remove from heat and whisk in the powdered sugar, 3 tablespoons of milk, and vanilla extract until silky and spreadable. Add the additional tablespoon of milk if needed for perfect consistency.
While the bars are still warm (but not hot), spread the frosting evenly over the surface. This allows some frosting to slightly melt into the top layer for that signature lunch lady bar quality.
Allow the bars to cool completely before cutting into squares.