In a large bowl, whisk together warm milk, yeast, egg, 1 Tbsp melted butter, sugar, and salt.
Add 3 cups flour and mix with a wooden spatula; add flour until dough is no longer sticky.
Knead dough 3 minutes by hand. Cover with plastic wrap; let rise 1 hour in a warm place.
Mix softened butter, brown sugar, and cinnamon to make the filling.
Dust work surface with flour; roll dough into 17 x 10 rectangle.
Spread filling evenly over dough, roll lengthwise, and cut into 8 rolls with dental floss.
Place rolls in 9x13 pan, cover with plastic wrap; let rise 30 minutes.
Brush rolls with remaining melted butter and bake at 350°F for 20-25 minutes.
Cool rolls about 15 minutes.
Beat softened butter, cream cheese, and vanilla until smooth.
Slowly add powdered sugar and mix until combined.
Spread frosting over warm cinnamon rolls and serve.