Combine 2 cups flour, sugar, dry yeast and salt in a large mixer bowl and stir until blended. Place water and butter in a microwave-safe bowl. Microwave on HIGH in 15 second increments until very warm but not hot to the touch (120° to 130°F, butter won’t melt completely). Add to flour mixture with egg.
Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add 1 cup flour; beat 2 minutes at high speed, scraping bowl occasionally. Stir in just enough remaining flour so that the dough will form into a ball.
Knead on lightly floured surface until smooth and elastic and dough springs back when lightly pressed with 2 fingers, about 6 to 8 minutes. Cover with a towel; let rest for 10 minutes.
While dough rests, remove core, peel and dice apples. Melt 1 tablespoon butter in medium skillet over medium heat. Add apples to cook for 4-5 minutes, stirring occasionally. Then add 2 tablespoons brown sugar and 1 tablespoon cinnamon. Stir until apples are coated and cooked. Set aside to cool. For remainder of the filling, combine sugar and remaining cinnamon (2 teaspoons) in a small bowl. Set aside.
Roll dough into a 15 x 10-inch rectangle using a rolling pin. Spread 3 tablespoons butter over dough stopping at least 1/2-inch from the edges on the long sides. Sprinkle with cinnamon sugar mixture. Then spread apples evenly on top. Beginning at long end of the rectangle, roll up tightly.
Pinch seams to seal. Cut into 12 equal pieces. TIP: Use unflavored dental floss instead of a knife to cut rolls. To do, cut a piece of floss about 12 inches long. Slide floss under the roll; bring the ends up and cross over to cut each slice. Place cut sides down, in greased 13 x 9 inch baking pan. Cover with towel; let rise in warm place until doubled in size, about 1 hour.
Bake in preheated 350ºF oven for 25 to 30 minutes or until rolls are golden brown. Cool on wire rack for at least 20 minutes.
Combine all frosting ingredients (start with 2 tablespoons milk and add more if needed) in a large bowl and beat until creamy. Drizzle or spread over rolls.
**If you are not using an electric mixer, just mix in a bowl 4 1/2 cups of flour, the sugar, salt, yeast, melted butter, egg, and 120º-130ºF water with a spatula until incorporated. Reserve the remaining 1/2 cup of flour for kneading time. Once the dough comes together, transfer to the counter and start kneading with your hands until the dough is smooth, elastic, and it springs right back when you lightly press it with your fingers. Kneading time by hand might be from 8-10 minutes.