Melt 4 Tablespoons of the butter in a small skillet over medium heat. Continue cooking until it becomes golden brown and fragrant, stirring often. Transfer to a heatproof bowl and place in the fridge to cool.
Move oven rack to the middle position and heat oven to 350°F. Line a 9x13" baking pan with foil and grease lightly with oil.
Mix 2 cups rolled oats, 1 cup all purpose flour, 1/2 cup whole wheat flour, 1/2 teaspoon salt, 1/2 teaspoon baking powder, and 1/4 teaspoon baking soda together in a medium bowl. Set aside.
When the butter has cooled to room temperature, add to the bowl of a stand mixer with the remaining 4 Tablespoons butter and 1 cup light brown sugar. Cream together on medium-high speed until light and fluffy. Add 2 large eggs, one at a time, beating in between each addition. Add 1 Tablespoon vanilla extract and beat until fluffy.
Turn off the mixer and add the oat mixture. Turn the mixer on to low speed and mix until combined. Mix in 1 1/2 cups butterscotch chips.
Spread the batter into the prepared pan, ensuring to spread the batter into the corners of the pan. Bake until golden brown, 25-30 minutes, rotating the pan halfway through baking time. Cool completely before cutting. Store at room temperature in an airtight container or freeze.