Prepare dough: Place milk, eggs, margarine, flour, salt, white sugar, and yeast in the pan of a bread machine in the order recommended by the manufacturer. Select dough cycle; press Start.
When dough has doubled in size, turn it out onto a lightly floured surface. Cover it with a kitchen towel or plastic wrap and let it rest for 10 minutes.
Roll dough on a lightly floured surface to a 16x21-inch rectangle.
Prepare filling: Combine brown sugar and cinnamon in a small bowl. Spread softened butter over the dough, then sprinkle cinnamon-sugar mixture evenly over top.
Starting at the longer end, roll up the dough; cut into 12 rolls. Place rolls in a lightly greased 9x13-inch baking pan. Cover and let rise until nearly doubled, about 30 minutes.
Meanwhile, preheat the oven to 400 degrees F (200 degrees C).
Bake rolls in the preheated oven until golden brown, about 15 minutes.