In a 2-quart pan over medium heat, heat milk to lukewarm. In a large bowl, dissolve yeast in 1 cup warm (110°) water. Let stand 5 minutes.
If using a heavy-duty mixer: With the paddle attachment, stir milk, 3 tablespoons sugar, 2 tablespoons butter, the salt, and 3 1/2 cups flour into yeast mixture. Beat on high until slightly stretchy, about 2 minutes. Switch to dough hook and, on medium speed, beat in 1 1/2 cups more flour until a stiff dough forms. Continue beating until dough pulls cleanly from bowl, 5 to 7 minutes longer; if dough is still sticky, add more flour, 1 tablespoon at a time.
If mixing by hand: With a wooden spoon, stir milk, 3 tablespoons sugar, 2 tablespoons butter, the salt, and 3 1/2 cups flour into yeast mixture. Stir vigorously until slightly stretchy, 2 to 4 minutes. Stir in 1 1/2 cups more flour until a stiff dough forms. Scrape onto a floured board and knead until smooth, elastic, and no longer sticky, about 8 minutes; add flour as required to prevent sticking. Return dough to bowl.
Cover bowl airtight and let dough rise at room temperature until doubled, 1 to 1 1/2 hours.
Scrape dough onto a floured board and press gently to expel air. Divide in half. Roll each half into a 10- by 16-inch rectangle. Spread 6 tablespoons butter over each rectangle. In a small bowl, mix 1 1/4 cups sugar with cinnamon. Sprinkle half the mixture over each rectangle.
Starting from a long edge, roll each rectangle into a tight cylinder. Cut each cylinder into six equal pieces. Place, cut side down and slightly apart, in two 7- by 11-inch baking pans. Cover pans and let stand at room temperature until rolls are almost doubled in size, 1 to 1 1/2 hours.
Bake in a 350° oven until rolls are browned, 30 to 35 minutes. Cool in pan 10 minutes, then drizzle with icing. Serve warm from the pan.
Icing: In a bowl, mix 3 cups powdered sugar, 1 teaspoon vanilla, and 1/4 cup water. Thin with water if icing is too thick to drizzle.