Preheat oven to 375°F and line a 13 x 9-inch baking dish with parchment paper.

Mix melted butter, sugar, and vanilla in a medium bowl with a spatula. Add flour and salt and stir until incorporated. Press dough evenly into bottom of prepared baking dish. Bake until crust is golden and slightly puffed, about 18 minutes. Transfer to a wire rack and cool crust in pan.

To prepare filling, cook butter in a heavy saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes. Pour browned butter into a glass measuring cup to cool slightly. Whisk together eggs, sugar, and salt in a medium bowl. Add flour and extracts and whisk until smooth. Gradually whisk in browned butter; whisk mixture until completely blended.

Arrange cherries over cooled crust and carefully pour filling evenly over fruit. Bake for about 30 minutes, until filling is puffed and golden, and a toothpick inserted in the center comes out clean. Cool bars completely in pan on a wire rack. Carefully lift parchment to remove bars from pan and place on a cutting board; cut into squares using a serrated knife. Store bars in an airtight container up to one day at room temperature, then store any remaining in the refrigerator.