Brown the butter: Melt the butter in a medium saucepan over medium heat. It'll foam, then settle, then turn amber. Swirl occasionally. The second you see brown specks and smell a nutty aroma, pull it off the heat. Don't walk away — it goes from brown to burned in about thirty seconds.
Cool the butter slightly: Pour the browned butter into your large mixing bowl. Let it sit for 5 minutes. Hot butter + raw egg = scrambled egg blondies, so be patient.
Whisk in the sugars, egg, yolk, and vanilla: Add both sugars and whisk until combined. It'll look like wet sand. Add the egg, egg yolk, and vanilla. Whisk vigorously for about a minute until the mixture is smooth, glossy, and slightly thickened.
Fold in the dry ingredients: Add the flour, baking powder, and salt. Fold gently with a spatula just until no streaks of flour remain. Overmixing makes tough blondies.
Prepare the berries: Toss the blueberries with 1 tablespoon of flour in a small bowl. This keeps them from sinking. Gently fold the floured berries into the batter. A few purple streaks are fine.
Spread into the pan: Scrape the batter into your prepared 8x8 pan lined with parchment. Spread evenly. If you want a bakery look, press a few extra berries on top.
Bake: Bake at 350°F for 28-32 minutes. The edges should be golden and pulling away from the pan. The center will look just barely set — that's perfect. It finishes cooking as it cools. The smell shift from raw batter to browned butter is your cue.
Cool completely before cutting: Let the pan cool on a wire rack for at least 30 minutes. If you cut them warm, they'll crumble. Use the parchment overhang to lift the slab out, then slice into 9 squares.