Prepare the Dough. In a bowl, combine warm milk and yeast. Let sit for 5 minutes until foamy. Add sugar, salt, melted butter, and egg. Mix until combined. Gradually add flour, mixing until a soft dough forms. Knead the dough on a floured surface for 8-10 minutes until smooth and elastic. Place the dough in a greased bowl, cover, and let rise in a warm place for 1-1.5 hours until doubled in size.
Make the Orange Filling. In a small bowl, mix softened butter, sugar, orange zest, and orange juice until well combined.
Shape the Rolls. Roll out the dough into a 12x18-inch rectangle. Spread the orange filling evenly over the dough. Roll up tightly from the long side and slice into 12 equal pieces. Place rolls in a greased 9x13-inch baking dish. Cover and let rise for 30-45 minutes.
Bake the Orange Rolls. Preheat oven to 350°F. Bake rolls for 20-25 minutes or until golden brown.
Make the Orange Glaze. Whisk together powdered sugar, orange juice, and vanilla extract. Drizzle over warm rolls before serving.