Wash and core the apples, then dice them into small bite-sized pieces so they cook quickly and evenly.
Add the diced apples to a large pot with a splash of water, the brown sugar, half the cinnamon, and nutmeg if using. Cook over medium heat for 6–8 minutes, stirring occasionally, until the apples soften and look glossy.
Pour in the oats, milk, and salt. Stir and bring to a gentle simmer.
Reduce the heat to low. Pour in the egg whites slowly while stirring continuously. Continue stirring 4–5 minutes until the oatmeal thickens and becomes creamy.
Remove from heat and stir in the remaining cinnamon. Taste and adjust sweetness if needed.