Pour the apple cider into a small saucepan and simmer until reduced to approximately 1/4 cup.
Set aside to cool slightly.
Place the butter in a light-colored saucepan over medium heat.
Cook, stirring frequently, until the butter foams and the milk solids become golden brown and nutty-smelling.
Remove from the heat and cool for approximately 10 minutes.
Preheat the oven to 350°F (175°C).
Line an 8 × 8-inch baking pan with parchment paper.
Whisk together the browned butter, brown sugar, granulated sugar, eggs, vanilla, and reduced apple cider.
Add the flour, cinnamon, and salt.
Fold gently until just combined.
Fold in the finely diced apple.
Transfer the batter to the prepared pan and spread evenly.
Bake for 28–34 minutes, until the edges are golden and the center is set but still soft.
Cool completely in the pan.
Lift out using the parchment paper and cut into 16 squares.