Preheat oven to 400 F / 200 C.
Spray two 8-inch round cake pans or one 9x13 baking dish with nonstick cooking spray.
Heat 1 ½ cups milk in the microwave until warm to the touch. Combine milk and 2 Tbsp. melted butter in the bowl of a standing mixer. Stir in yeast and 2 tablespoon sugar. Let sit for 5 minutes.
To the yeast mixture, add flour, and salt. Mix on low until the ingredients come together. Increase speed to medium and mix for 5 minutes, until the dough is smooth and elastic and begins to make a slapping sound as it hits the sides of the bowl.
Remove the bowl from the mixer and cover the dough with a dish towel. Let dough rise (preferably near the warm oven), until the surface is smooth, about 15 minutes.
Turn the dough out on a lightly floured surface and roll it into a rectangle (approximately 9"x18") with the long side facing you.
Spread 3 Tbsp. softened butter over the dough. Combine ¼ cup sugar and cinnamon. Sprinkle sugar-cinnamon mixture over softened butter.
Starting with the side closest to you, roll up the dough. Slice roll of dough in half. Slice each half into 9 rounds (you should have 18 rolls total). Place the slices in the prepared baking dish(es) and let rise uncovered in a warm place (preferably near the warm oven) until the edges of the rolls are touching, 15 minutes.
Bake until light golden brown on top, 15 to 18 minutes.
While the rolls are baking, use a hand mixer to beat the cream cheese, powdered sugar, and vanilla. While beating, add milk a little at a time, until the mixture is smooth and pourable.
When the rolls come out of the oven, let them cool for 5 minutes before drizzling icing over top.
Serve immediately (of course...who could wait!).