unsalted butter, at room temperature; half can be melted and then combined with the unmelted half, light brown sugar, Kosher salt (1/2 teaspoon fine salt), maple syrup (robust type, if possible), large egg, vanilla extract, maple extract, if available (otherwise, vanilla type will do), all-purpose flour, baking soda, maple sugar or maple flakes (if not available, use maple syrup), (about 7 ounces) walnuts, toasted and chopped, divided