Start by preheating your oven to 350 degrees F. While it heats, spray a 9x13 inch baking dish generously with cooking spray so nothing sticks once it's baked. Set the dish aside on the counter so it's ready the moment your batter is mixed.
In a large bowl, whisk the eggs, oil, and sugar together until the sugar dissolves into the mixture. Add the milk, applesauce, 2 teaspoons of vanilla, the cinnamon, and the salt, then whisk again until everything looks smooth and well combined. This forms the base that carries all the flavor through the bake.
Add the oats, sliced strawberries, and baking powder directly into the wet mixture. Stir until every oat looks coated in the liquid, since dry patches can bake up chewy instead of tender. Pour this mixture into your prepared baking dish and spread it into an even layer with a spatula.
In a small bowl, combine the softened cream cheese, powdered sugar, 1 tablespoon of milk, and the remaining teaspoon of vanilla. Whisk until it's smooth and thin enough to pour slowly off a spoon. If it feels too thick, add a splash more milk or warm it in the microwave for a few seconds.
Drizzle the cheesecake mixture back and forth over the oats in a zig-zag pattern. Turn the dish a quarter turn, then drag a butter knife through the batter in the opposite direction to create a marbled swirl. Bake for 40 minutes, then let it cool for at least 10 minutes before slicing.