In a large mixing bowl, combine all-purpose flour, sugar, yeast, and salt. In a separate bowl, whisk together warm milk, melted butter, and an egg, then gradually incorporate this mixture into the dry ingredients. Knead until the dough is smooth and elastic, about 5-7 minutes, then cover and let it rise in a warm place until doubled in size—approximately 1 hour.

While the dough is rising, peel and dice your Granny Smith apples. In a medium saucepan over medium heat, combine the diced apples, brown sugar, cinnamon, and lemon juice. Cook for 5-7 minutes until tender and fragrant. Remove from heat and set aside to cool.

In a medium bowl, use an electric mixer to beat cream cheese, powdered sugar, and vanilla extract until smooth and creamy, about 2 minutes. Ensure no lumps remain.

Once the dough has risen, punch it down and roll it into a rectangle about ¼ inch thick. Spread the cream cheese mixture over the surface, followed by the cooled apple filling. Roll tightly into a log, then slice into equal pieces, about 1½ inches thick.

Mix oats, brown sugar, flour, and cold diced butter in a bowl until it resembles coarse crumbs. Place the sliced buns in a greased baking dish and sprinkle with the crumble topping.

Preheat your oven to 350°F (175°C). Bake for 25-30 minutes, until golden brown and bubbly. Monitor closely to avoid over-browning.

After removing from the oven, cool on a wire rack for about 10 minutes. Whisk powdered sugar, milk, and vanilla until smooth and drizzle over warm buns.