Cake:
1/2 cup unsalted butter (1 stick), cut into pieces
1/2 cup water
2 cups all-purpose flour
1 teaspoon baking powder
1/4 teaspoon baking soda
1/4 teaspoon salt
1/2 teaspoon cinnamon
1/4 teaspoon grated nutmeg
1 1/2 cups sugar
2 eggs, beaten
1/2 cup reduced-fat sour cream (NOT fat-free)
1/2 cup unsweetened applesauce
1 teaspoon vanilla extract
2 medium apples, peeled and grated
Icing:
1 cup brown sugar, light or dark or a combination
2 tablespoons butter
6 tablespoons (1/4 cup plus 2 tablespoons) milk or half-and-half, plus additional as needed
2 cups confectioner's (powdered) sugar
1/2 teaspoon vanilla extract
What to Do:
1. Round up a sheet pan approximately 9" by 13" in size. You can fudge a little on these measurements...7x11 or 10x15, for example. But don't try to cram this quantity of batter into a 9-inch square or stretch it across a half-sheet pan. Butter the pan or spray it with nonstick cooking spay.
2. Preheat your oven to 375 degrees (yes) and make sure your rack is in the middle position.
3. In a small bowl, stir together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg with a fork or whisk. Set aside.
4. In a large saucepan, bring the 1/2 cup (1 stick) butter and the water just to a full boil, stirring once or twice while waiting for this to happen.
5. As soon as the butter/water mixture boils, pull the pan off the heat. Dump the flour mixture into the pan, followed by all the remaining cake ingredients in the order listed. Stir everything together with a wooden spoon just until everyone is in the party and you don't see any rebellious pockets of dry flour mixture.
6. Pour the cake batter into the prepared pan, smooth the top, and bake for 15-20 minutes or until a toothpick inserted into the center of the cake emerges with moist crumbs clinging to it. Be careful not to overbake this cake. You want the aforementioned moist crumbs—not wet batter, but not a totally clean pick, either. A minute can make a difference here, so start checking early and keep checking often. As my mom always says, you can add cooking time, but you can't take it away. Remove pan from oven and set on a wire rack to cool for 10 minutes.
7. When your cake has cooled for 10 minutes, make your icing by melting the 1 cup brown sugar, the 2 tablespoons of butter, and the milk or half-and-half in a medium-sized saucepan over medium heat. Bring the mixture to a boil, stirring constantly, then reduce the heat to medium-low and cook (keep stirring!) until the sugar is dissolved. Remove from heat and let cool for 3 minutes, then add the confectioner's sugar and the vanilla extract and beat with a wooden spoon until smooth and/or you have worked out all your frustrations, whichever comes first. If the icing looks too thick to pour easily (it will set up quickly once you pour it on the cake, so you don't want it too thick at this stage), add a little more milk or (even better) half-and-half, a few drops at a time, until you have a pourable but not runny consistency.
8. Immediately pour the warm icing over your warm cake and spread it evenly and without delay from edge to edge. (This will be much easier if you don't pour all the icing in one spot in the middle of your cake but rather bake and forth from end to end, nearly covering the entire surface just in the pouring.)
9. Allow the icing to set up for at least 20 minutes, then enjoy your cake warm or at room temperature. If you have any to store, be sure to cover your pan tightly with plastic wrap or foil so it doesn't dry out. Eating the entire cake the day it's made also solves this problem nicely. But I will say that this cake keeps well. I'll leave it to you to work out that paradox.
Found at https://guiltychocoholicmama.blogspot.com/2018/10/autumn-in-sheet-pan-caramel-apple-cake.html