Preheat oven to 350°F. Grease a 9X13 inch pan or line with parchment paper.
Place 1/2 cup of white chocolate chips in a small bowl.
To make the brown butter*: Place the butter in a saucepan and melt over medium heat. Continue to cook stirring occasionally as the butter starts to bubble and foam. After a few minutes the foam will subside and the butter will be a dark golden color and have a nutty, caramelly fragrance. It is finished when you see tiny brown flecks on the bottom of the pan.
Immediately pour the hot brown butter over the white chocolate chips, scraping the pan with a rubber spatula to include the brown bits. Stir until chips are melted and mixture is smooth. Allow to cool slightly
In a medium bowl, beat the eggs, both sugars, milk, and vanilla until smooth. Add the brown butter and chocolate mixture and stir to combine.
Add the flour and salt and stir until just combined. Do not over mix. Pour into prepared pan and smooth with a rubber or offset spatula. Sprinkle top with remaining white chocolate chips and optional sprinkles.*
Bake for 30-35 minutes until golden and an inserted toothpick comes out clean. Allow to cool before slicing.