Preheat the oven to 350°F and grease a 9x13-inch baking pan. Stir the almond milk and apple cider vinegar together and set aside for 5 minutes.
In a medium bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, and nutmeg.
In a large bowl, whisk the melted butter, brown sugar, and granulated sugar until smooth. Whisk in the eggs, vanilla, and almond milk mixture until well combined.
Stir the dry ingredients into the wet ingredients with a spatula just until combined, then fold in the diced apples. Spread the thick batter evenly into the prepared pan.
Bake for 38 to 45 minutes, until golden and a toothpick inserted in the center comes out clean. Let the cake cool completely in the pan on a wire rack, about 1 1/2 hours.
For the frosting, melt the butter in a saucepan over medium heat, stir in the brown sugar, and bring to a boil, stirring constantly, for 2 minutes. Carefully stir in the almond milk, return to a boil, then remove from the heat and cool for 10 minutes.
Beat in the powdered sugar and vanilla with a hand mixer until thick, smooth, and spreadable. Immediately spread the caramel frosting over the cooled cake, sprinkle with flaky sea salt, and let it set for 20 minutes before slicing.