Warm apple cider and whole milk to exactly 110°F in a small saucepan. Pour into stand mixer bowl, sprinkle with yeast and 1 teaspoon sugar, stir gently, and let sit 5-10 minutes until foamy.
Add remaining sugar, melted butter, and egg to yeast mixture; beat on low for 30 seconds. Whisk flour and salt together, gradually add to mixer, then knead on medium for 6-8 minutes until soft and slightly tacky.
Grease a large bowl, place dough inside, turning once to coat. Cover tightly and let rise in a warm spot for 1 hour, until doubled in size and the dough passes the poke test.
Stir together brown sugar and cinnamon for the filling; set aside. Line a 9x13-inch baking dish with parchment paper, letting it hang over the edges, and soften the 1/2 cup butter for the filling.
Punch down risen dough and roll into a 16x12-inch rectangle, about 1/4 inch thick. Spread softened butter evenly, leaving a 1/2-inch border on one long edge, then sprinkle cinnamon-sugar mixture over the butter and press gently.
Roll dough tightly into a log starting from the long edge opposite the bare border, then pinch the seam to seal. Cut into 12 even rolls using a sharp knife or dental floss and place cut-side up in the prepared baking dish.
Cover pan loosely and let rolls rise for 30-45 minutes until puffy and touching. Meanwhile, preheat oven to 350°F.
Bake rolls for 25-30 minutes, rotating the pan halfway through, until tops are golden brown and centers feel firm. Cool in the pan on a rack for 10 minutes.
Whisk powdered sugar, apple cider, vanilla, and heavy cream until smooth and pourable, adjusting consistency as needed. Drizzle generously over warm rolls.
Lift rolls out of the pan using the parchment handles and serve warm.