Add the milk (warmed to about 98–105°F), eggs, softened butter, flour, salt, sugar, and rapid-rise yeast to a large bowl or the bowl of a stand mixer. Stir until the mixture starts to form a cohesive dough.

Knead the dough until smooth: about 5–8 minutes by hand on a lightly oiled surface or 4–7 minutes in a stand mixer fitted with a dough hook on medium speed.

Place the kneaded dough in a lightly oiled bowl, turn it once to coat, cover with plastic wrap, and let it rise in a warm spot until doubled in size, roughly 60 minutes.

Punch the risen dough down and transfer it to a lightly greased surface. Roll it into a 16x21-inch rectangle.

Spread the softened filling butter evenly across the dough, leaving a 1-inch strip along one short edge uncovered.

Evenly sprinkle the packed brown sugar and ground cinnamon over the buttered area, bringing the filling to the edges except for the reserved 1-inch border.

Starting from the opposite short end (the edge with filling), roll the dough tightly into a log, sealing the bare border along the seam.

Slice the log into 12 equal rolls and arrange them, cut side up, in a lightly greased 9x13-inch baking pan.

Cover the pan with plastic wrap and let the rolls rise again until nearly doubled, about 30 minutes.

While they finish rising, preheat the oven to 400°F (200°C).

Bake the rolls until the tops are golden brown, about 15 minutes. Remove from the oven and let cool for 5–10 minutes before icing.

In a small bowl, beat the room-temperature cream cheese and butter until smooth. Add the confectioners' sugar and vanilla, and continue beating until the frosting is creamy and lump-free.

Spread the icing over the warm (not piping hot) rolls so it melts slightly into the swirls. Serve warm.