Whisk the flour, yeast, granulated sugar, and salt in a large bowl.
Stir in the warm milk, egg, and melted butter until a shaggy dough forms, then knead for 5 minutes until smooth.
Cover and let the dough rise in a warm place until puffy, about 45 minutes.
Stir the caramel base ingredients together and spread them in a greased 9-inch round pan.
Roll the dough into a 12-by-9-inch rectangle, spread with softened butter, and sprinkle with brown sugar and cinnamon.
Roll tightly from the long side and cut into 12 pieces. Arrange them over the caramel base.
Let the rolls rest 15 minutes while the oven heats to 350 F.
Bake 18 minutes, until golden and the centers look set.
Cool 5 minutes, invert onto a plate, and serve warm.