Preheat oven to 350°F. Line or generously grease an 18 x 13-inch or 17 x 12.25-inch jelly roll pan. Set aside.
Beat butter and brown sugar in a large bowl until smooth and creamy, like peanut butter. Add eggs and vanilla and mix until combined. Mix in baking soda, salt, and oats until just combined. Add peanut butter and mix again. Add flour and mix just until combined.
Press dough into an even layer in the prepared pan. Bake for 15 to 17 minutes. Do not over-bake. (The bars should be soft for the best texture.)
Let bars cool for 5 minutes. Warm the peanut butter in microwave until pourable, then drizzle over the bars. Use a knife or spatula to gently and carefully spread into an even layer. Take your time here. Let bars cool completely until the peanut butter sets. Chill if needed to speed this up.
Add all frosting ingredients to the bowl of a stand mixer. Beat with whisk attachment until smooth and creamy. Spoon frosting over the bars and spread into an even layer.
Run a sharp knife under hot water, dry it, then cut into bars for clean slices.
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Notes
Note 1: Use old-fashioned oats. Quick oats absorb more liquid and will make the bars dry.Note 2: Use creamy peanut butter (I recommend Skippy). Skip natural peanut butter that separates.Note 3: A 17 x 12.25-inch or 13 x 13-inch pan is key for the right thickness.Note 4: Butter should be soft enough to press with your finger but not greasy or melted.Storage: Store bars covered at room temp for 5 days. To freeze, wrap each bar and store in a freezer-safe bag for up to 3 months. Thaw overnight in the fridge.