When I was a kid, my family always went apple-picking in the fall. Then we'd make a big bundt cake, with a recipe similar to this one. I adapted the recipe for the microwave so I could enjoy it more often.
Since it uses a whole apple, this is a pretty big cake, more of a “bowl cake” than a “mug cake.” I make it either in a small glass baking dish or in two mugs.
This turns out more like itself than, say, yellow cake does when translated to the microwave, perhaps because the apple is the focus and you can’t mess up an apple by imprecise measurements.
Peel and chop a large apple, and place it in a medium mixing bowl. Add 3 tablespoons plus 1 teaspoon of packed brown sugar, ½ teaspoon cinnamon, a pinch of salt, a splash of vanilla extract, 2 tablespoons vegetable oil, about half an egg,* ½ cup all-purpose flour, and ½ teaspoon baking soda.
Stir well. It will start off very stiff, but as you stir, some juice from the apple will make it more reasonable.
Butter or spray the microwaveable baking dish(es) of your choice: this will make two single-serving cakes quite nicely, or one cake from which you can slice servings of custom size. Scrape the batter into the prepared dishes.
Microwave at 100% power. Depending on your microwave, this can take up to three and a half minutes. That sounds long for a mug cake, but you need to cook the apple, after all.
Let cool in the dish for half a minute, then turn out onto plates.
*Dividing an egg in half isn’t the easiest thing ever. It doesn’t have to be all that exact, though; if it's scant, it'll be more biscuity, and if it's generous it'll be more custardy. Neither is bad. And you can just double the recipe and make four cakes, because the apple means this keeps well and stays moist.