In a bowl, whisk together the eggs, milk, applesauce, oil, and vanilla.
In another, place the flour, sugar, baking powder, cinnamon, salt, and nutmeg and combine.
Transfer the mixture to a large bowl and stir in the apple. Add the egg mixture and use a spatula to fold until the dough just comes together. Be careful not to overmix.
Turn the dough out onto a floured surface. Knead lightly, using more flour as needed to prevent sticking. Form the dough into a ball, divide it in half, and flatten each half into a 1-inch-thick disk. Place on a parchment-lined baking sheet and freeze for 20 minutes.
Preheat the oven to 400°F and line another baking sheet with parchment paper.
Remove the dough from the freezer and cut each disk into 6 wedges. Separate the wedges and arrange them on the prepared baking sheet. Bake for 15 to 17 minutes, or until golden brown around the edges. Transfer to a wire rack to cool.
For the Glaze
In a bowl, whisk together the powdered sugar, milk, and vanilla until smooth. Drizzle over the cooled scones.