In a skillet over medium heat, simmer the apple cider until it reduces to about half a cup, roughly 10-15 minutes. Let it cool slightly before using. This step concentrates the apple flavor so it doesn't get lost in the dough.
In a large bowl, whisk together 2½ cups of flour, sugar, salt, and instant yeast. Mixing the yeast directly with the dry ingredients keeps everything evenly distributed. Set this bowl aside while you prep the wet ingredients.
Combine the reduced cider, milk, and butter in a microwave-safe bowl and heat until it reaches 115-120°F. Stir in the nutmeg once warmed. It should feel hot but never boiling, since overheated liquid can kill the yeast.
Pour the warm liquid over the dry ingredients and stir to combine. Add the egg and remaining flour, mixing until a soft dough forms. The dough is ready when it turns elastic and starts pulling away from the bowl's sides.
Turn the dough onto a floured surface and knead gently for 3-4 minutes until smooth. Place it in a greased bowl and let it rest for 10 minutes. This short rest relaxes the gluten, making the dough easier to roll out later.
Shape the dough into a rectangle, then roll it out to about 8x14 inches and ¼-inch thick. Dust with flour as needed if it gets sticky. Using a ruler helps keep your rolls evenly sized, though it's fine if they're not perfect.
Spread softened butter over the dough, followed by a layer of apple butter, leaving a half-inch border. Mix the brown sugar and cinnamon together, then sprinkle it evenly across the top. This double layer is what makes the filling so rich.
Starting from a long edge, roll the dough tightly into a log. Use a sharp serrated knife or dental floss to slice it into 9-12 even rolls. Floss gives you a cleaner cut without squashing the delicate layers.
Arrange the rolls in a greased 9-inch pan, leaving a little space between each. Cover with plastic wrap and let rise in a warm spot for 50-75 minutes, until roughly doubled. I like using my microwave (turned off) as a cozy, draft-free spot.
Preheat your oven to 375°F and bake the rolls for 20-25 minutes, until lightly browned and cooked through. Cover loosely with foil if they start browning too fast on top.
Melt butter in a saucepan, whisking until it turns golden brown and smells nutty. Whisk in powdered sugar and vanilla, then add milk until you reach a pourable consistency. Drizzle generously over the warm rolls before serving.