In a small saucepan, combine the apple cider and milk over medium heat until warm (not boiling). Remove from heat and whisk in the melted butter and sugar until dissolved. Sprinkle yeast over the mixture, let it sit for about 5-10 minutes until it becomes foamy.

In a large mixing bowl, combine the flour, salt, and cinnamon. Make a well in the center and pour in the yeast mixture along with the vanilla extract.

Mix until fully combined and then knead the dough for about 5-7 minutes on a floured surface until it's smooth and elastic. Place it in a greased bowl, cover it with a cloth, and let it rise in a warm place for about 1 hour, or until it doubles in size.

In a bowl, combine the brown sugar and cinnamon. Once the dough has risen, roll it out on a floured surface into a rectangle (about 1/4 inch thick). Spread melted butter over the surface and sprinkle the cinnamon-sugar mixture evenly.

Roll the dough tightly from one side to another into a log. Slice the log into 12 equal pieces and place them in a greased baking dish, leaving some space for them to expand.

Cover the rolls with a cloth and let them rise again for about 30 minutes.

Preheat the oven to 350°F (175°C). Bake the rolls for 20-25 minutes until golden brown.

Whisk icing sugar and milk together until you get the desired glaze consistency. Drizzle over warm rolls before serving.