Preheat the oven to 350ºF degrees. Line an 8-inch square baking dish with parchment paper and spray with nonstick cooking spray; set aside.
In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma.
Once browned and nutty, turn off the heat and immediately pour into a large mixing bowl. Allow to cool for 5 minutes.
Whisk the sugar, eggs, and vanilla into the brown butter until smooth. Switch to a wooden spoon, add the flour, baking powder and salt, and stir until just combined.
Pour half of the batter into the baking pan. Dollop half of the filling evenly on top, then swirl with a knife. Pour the rest of the batter on top, followed by dollops of the rest of the filling, and swirl once again with a knife.
Bake for about 35-40 minutes, or until edges are golden brown and the top of the blondies looks set. The center will still be jiggly, but the blondies will continue to bake as they cool. Allow blondies to cool completely before cutting into squares.