Heat your oven to 350 degrees F. Line an 8x8 baking pan with parchment paper, leaving a couple inches of overhang on two sides so you can lift the bars out easily. Lightly grease the parchment.

Brown the butter: Place the butter in a light-colored medium saucepan over medium heat. Stir constantly as it melts, foams, and then begins to smell nutty. You will see golden brown bits forming on the bottom of the pan. Pull it off the heat the moment it smells like toasted hazelnuts and turns a deep amber color. This takes about 5 to 6 minutes. Pour it immediately into your large mixing bowl so it stops cooking.

While the butter cools for about 3 minutes, toss your diced apple pieces with the lemon juice in a small bowl and set aside.

Whisk the brown sugar into the warm brown butter until fully combined. Add the eggs one at a time, whisking well after each. Add the vanilla and whisk again. The batter should look glossy and smooth.

Add the flour, cinnamon, baking powder, and salt directly to the bowl. Switch to a rubber spatula and fold until just combined, stopping as soon as you no longer see dry streaks. Do not overmix.

Fold the apple pieces into the batter gently. The batter will be thick. Spread it evenly into the prepared pan, smoothing the top with your spatula.

Bake on the center rack for 28 to 32 minutes. The top should look crackled, dry, and golden. When you press the center lightly with your fingertip, it should have just a slight give but not feel wet or jiggly. A toothpick inserted near the center should come out with a few moist crumbs, not raw batter.

Let the pan cool completely on a wire rack before lifting the bars out by the parchment. For the cleanest cuts, refrigerate the whole slab for 30 minutes, then slice into 16 squares with a sharp knife.