Warm water and milk until comfortably warm, add egg, and whisk until combined.

Pour wet mixture into dry ingredients, stir with a wooden spoon until a thick, sticky dough forms.

Gradually add additional flour while mixing until dough pulls together into a slightly tacky ball.

Turn dough onto a floured surface and knead 5–7 minutes until smooth and elastic.

Cover and let dough rest 20 minutes to relax gluten, making it easier to roll.

Roll out dough into a rectangle roughly 14×10 inches with a lightly floured rolling pin.

Spread softened butter evenly over dough surface.

Sprinkle brown sugar and cinnamon over butter, leaving a small border around the edges.

Starting from one long side, tightly roll dough into a log.

Trim ends and slice log into 12 even pieces using a serrated knife or dental floss.

Arrange rolls in a greased 9×13-inch pan and cover; let rise until doubled in size.

Preheat oven to moderate heat, bake rolls until golden and puffed.

In a bowl, beat cream cheese, butter, powdered sugar, and vanilla until smooth.

Spread frosting over warm rolls so it seeps into the swirls