Activate the yeast: Mix warm milk with sugar in a small bowl. Sprinkle yeast over the mixture and let it sit for 5-10 minutes until frothy. Make the dough: In a large bowl, combine flour and salt. Add the yeast mixture, melted butter, and vanilla extract. Stir until a dough forms.
Knead the dough: Turn the dough onto a floured surface and knead for 5-7 minutes until smooth and elastic. Cover the dough and let it rise for 30 minutes.
Prepare the filling: Mix brown sugar and cinnamon in a small bowl. Spread softened butter over the rolled-out dough, then sprinkle with the cinnamon-sugar mixture.
Roll and slice: Roll the dough into a log and cut it into 12 equal pieces. Place the rolls in a greased 9x13-inch baking dish. Bake: Preheat the oven to 350°F (175°C). Let the rolls rise for 10-15 minutes, then bake for 20-25 minutes or until golden brown.
Make the glaze: Whisk together powdered sugar and milk to make a simple glaze. Drizzle over the baked cinnamon rolls.
Serve: Serve warm and enjoy the fluffy, egg-free cinnamon buns!
