Warm 1 cup milk to about 110°F
Stir in 2 1/4 teaspoons active dry yeast and 1 tablespoon sugar
Let sit for 5 to 10 minutes until foamy
Mix 1/4 cup melted butter, 1/4 cup sugar, 1 egg, and 1/2 teaspoon salt
Add the yeast mixture
Gradually add 3 to 3 1/2 cups all-purpose flour
Knead until smooth and elastic
Place dough in a greased bowl
Cover and let rise until doubled, about 1 to 1 1/2 hours
Roll dough into a rectangle
Spread with 1/4 cup softened butter
Sprinkle with 1/2 cup brown sugar and 2 tablespoons cinnamon
Roll up tightly from the long side
Cut into 12 rolls
Place rolls in a greased baking dish
Cover and let rise until puffy, about 30 to 45 minutes
Bake at 350°F for 20 to 25 minutes
Mix 1 cup powdered sugar, 2 tablespoons milk, and 1/2 teaspoon vanilla
Drizzle icing over warm rolls