Stir 2½ teaspoons of active dry yeast and ½ teaspoon of granulated sugar into ½ cup of warm water and leave until foamy.
In a large bowl, mix ½ cup of warm whole milk, ¼ cup of granulated sugar, ⅓ cup of melted butter, 1 teaspoon of salt, 1 large egg, and the yeast mixture.
Gradually mix in 3½ cups of all-purpose flour until the dough pulls away from the sides of the bowl. Knead for 5 minutes, then cover and let rise for about 1½ hours, until doubled.
Roll the dough into a 20 × 18in / 50 × 45cm rectangle. Spread with 6 tablespoons of softened butter and sprinkle with ⅔ cup of packed light brown sugar mixed with 2 tablespoons of ground cinnamon.
Roll up tightly and cut into 12 rolls. Place in a buttered 9 × 13in / 23 × 33cm pan, cover, and let rise for about 1 hour, until doubled. Toward the end, preheat the oven to 375°F / 190°C.
Bake for 18-22 minutes, until no longer doughy in the center.
For the icing, beat 4 tablespoons of softened butter with 2 cups of icing sugar. Beat in 1 teaspoon of vanilla extract and 3 tablespoons of milk, adding more milk until spreadable, then spread over the rolls.
