Preheat the oven to 325 degrees Fahrenheit with a rack in the center position. Grease a sheet pan with butter.

In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg.

In a large bowl, whisk together the brown sugar, melted butter, and oil until smooth. Add the eggs, vanilla, and almond extract and whisk well to combine.

Add the dry ingredients to the wet ingredients and stir with a rubber spatula until the dough just comes together. Fold in the apples until evenly distributed.

Pour the batter into the prepared pan and spread it evenly to the corners with a large offset spatula. Bake for 20 to 25 minutes, until the cake is deeply golden and the edges just start to pull away from the sides of the pan. Set the sheet pan on a wire rack and let the cake cool for 20 to 25 minutes.

Melt the butter in a small pot over medium heat. Cook, stirring often with a rubber spatula, until the butter turns a toasted, amber color (it will bubble and spit; when it quiets down, check for browning). Whisk in the confectioners’ sugar, maple syrup, cinnamon, salt, and milk until smooth.

Drizzle the glaze evenly over the cooled cake. Let set for five minutes.

Slice the cake into pieces and serve. The cake is best the day it’s made, but will keep, tightly covered, in the refrigerator for three to four days.