To make the dough, combine the milk, butter, and 1/2 cup water in a medium saucepan. Set over medium-low heat and cook until just warmed through and the butter has almost melted completely. Set aside to cool until warm. Pour into the bowl of a stand mixer and sprinkle in a pinch of sugar, and the yeast. Let stand until foamy, about 10 minutes.

Add in the egg yolk, vanilla, sugar, flour, salt and cinnamon. Mix on low with the dough hook until the mixture comes together. Increase the speed to medium and knead until the dough is soft, about 6 minutes. Grease a large bowl with butter, oil or cooking spray. Add the dough to the greased bowl, turning over to coat both sides. Cover with plastic wrap and a damp kitchen towel and place in a warm draft-free spot to rest until doubled in size, about 1 hour.

In the meantime, for the filling, set a skillet over medium-high heat with 2 tablespoons butter, once melted. Add the sliced apples and sauté for about 2 minutes to soften them a bit. Sprinkle 1/2 teaspoon cinnamon, nutmeg, ginger, cardamom, and brown sugar. Cook, stirring often, until the apples have softened even more, about 10 minutes. Remove from heat and allow to cool completely. In a small bowl, combine the granulated sugar with the remaining 1/2 teaspoon cinnamon.

Punch the dough down into the bowl, and then transfer to a well-floured work surface. Roll out into a large rectangle about 1/4-inch thick. Spread the remaining 2 tablespoons butter over the dough and sprinkle with the cinnamon sugar. Then add the sautéed apples on top in an even layer, only adding in a little of the caramel. You don't want to add all of it because then you'll end up with soggy buns. Roll the dough into a tight log, starting from the long side. Trim off the ends and then cut into 12 even pieces. Place each piece into a well-greased muffin tin. Cover with plastic and a damp cloth and let rest in a warm draft-free spot until doubled in size, about 1 hour.

Preheat oven to 350 degrees F.

Bake the muffins until golden brown and fully cooked through, about 25 to 30 minutes. Remove from the oven and let cool for about 10 minutes in the pan before inverting them.

To make the glaze, in a small bowl, whisk together the powdered sugar, butter, vanilla, salt and enough milk to make a spreadable glaze. Spread the mixture over the warm muffins and serve immediately. Muffins can be wrapped in plastic and stored at room temperature for up to 2 days. They can also be frozen (without the glaze) for up to 3 months. Just thaw out and rewarm before serving. Enjoy!