Brown the Butter:
Place butter in a light-colored saucepan over medium heat. Stir occasionally as it melts and foams.
Watch carefully as butter turns golden brown with fragrant brown bits at the bottom (5-7 minutes).
Remove from heat immediately and let cool for 10 minutes.
Prepare and Mix:
Preheat oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment paper, leaving overhang on sides.
In a large bowl, combine oats, flour, brown sugar, baking soda, and salt. Whisk until evenly distributed.
Pour cooled browned butter into dry ingredients. Add vanilla extract and eggs. Stir until just combined.
Fold in milk chocolate chips and semi-sweet chocolate chips until evenly distributed.
Bake:
Spread mixture evenly in prepared pan, pressing down firmly with back of a spoon.
Bake for 15 minutes, or until edges are golden and center looks set.
Let bars cool completely in pan (about 1 hour). Use parchment overhang to lift out and cut into squares.