In a small pot over medium heat, melt 1 cup plus 2 tablespoons butter. Continue to cook, stirring or swirling constantly, until milk solids in bottom of pot are deeply golden brown, 4 to 6 minutes. (Butter will foam, so if needed, lift solids with a spoon from bottom of pot and check.)

Pour browned butter into a large heatproof bowl, making sure to scrape browned solids from bottom and sides of pot. Let cool to room temperature, about 10 minutes.

Meanwhile, in a large skillet over medium heat, melt remaining 2 tablespoons butter. Add apples and cook, stirring frequently, until fork-tender, 4 to 5 minutes. Let cool.

Preheat oven to 350°. Spray a 13" x 9" metal baking pan with cooking spray and line with parchment, leaving an overhang on 2 long opposite sides.

Transfer 2 tablespoons brown butter to a small bowl; set aside. To remaining brown butter, add brown sugar, 2 teaspoons vanilla, and 1 teaspoon salt and whisk until smooth. Whisk in eggs, then add baking soda and whisk just until distributed. Using a rubber spatula, fold in flour just until combined. Fold in apples. Dollop dough into prepared pan; smooth in an even layer (it will be stiff).

Bake blondies until just brown around the edges and deep golden on top, 20 to 25 minutes. Let cool in pan 10 minutes. Using parchment overhang, lift blondies out of pan to a wire rack and let cool completely.

To bowl with reserved brown butter, add confectioners’ sugar, milk, a pinch of salt, and remaining ¼ teaspoon vanilla. Whisk until glaze is a thick, pourable consistency.

Drizzle glaze over cooled blondies. Let set to let glaze harden, about 5 minutes, then cut into bars.