Preheat oven to 350°F. Grease or line a 9x13 baking pan with parchment.
Beat softened butter, granulated sugar, and brown sugar on medium speed for 2 to 3 minutes until light and fluffy. Add peanut butter and beat to combine.
Add eggs one at a time, mixing after each. Add vanilla and mix briefly.
Whisk together flour, oats, baking soda, and salt in a separate bowl. Add to wet ingredients and mix on low until just combined.
Press dough evenly into the prepared pan. Bake 18 to 22 minutes until set at the edges and barely underdone in the center. Cool completely.
Beat frosting ingredients together, adding milk one tablespoon at a time until thick and spreadable. Spread evenly over cooled bars.
Let frosting set 20 minutes at room temperature. Slice into rectangles and serve.