In a large bowl, mix warm milk, yeast, and 1/2 tbsp sugar. Let sit for 5 minutes until frothy.
Stir in the granulated sugar, oil, vanilla, melted butter, and egg.
Gradually mix in the flour and salt until a dough forms.
Lightly dust the counter with flour and knead the dough for 6-8 minutes, adding a bit more flour if too sticky.
Cover and let rise in a warm place for 35 minutes.
Lightly flour the counter, deflate the dough, and roll it into a rectangle about 9×12 inches.
Spread softened butter over the dough, then sprinkle with cinnamon and brown sugar. Press down gently to ensure it sticks.
Cut into six long strips and roll each one up tightly, tucking the ends under.
Place rolls in an 8-inch cake pan or small baking dish, ensuring they are close but not squished.
Preheat the oven to 350°F and let the rolls rise on the stovetop for 15 minutes while the oven heats.
Pour 2-3 tbsp of heavy cream or milk over the rolls before baking to enhance their softness and richness.
Bake for 25 minutes until golden brown and fluffy.
While baking, whisk together melted butter, powdered sugar, vanilla, and milk for the icing. Adjust consistency if needed.
Remove rolls from the oven, drizzle generously with icing, and enjoy warm!