Combine 1 cup 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.
Beat 2 minutes at medium speed of electric mixer, add egg and mix, 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 about 6 to 8 minutes. Dough should spring back when lightly pressed with 2 fingers. Cover with a towel; let rest for 10 minutes.
While dough is resting, prepare caramel syrup. Combine butter, brown sugar, and corn syrup in a 10-inch iron (ovenproof) skillet. Heat on stovetop until butter melts completely. Stir until smooth. Remove from heat and set aside to cool slightly.
Roll dough to a 12 x 15-inch rectangle. For filling, mix sugar and cinnamon in a small bowl. Spread the softened butter over dough and sprinkle cinnamon sugar evenly over butter. Roll up from long side, inching seam to seal. Cut into 10 equal slices. Arrange rolls, cut side down, on top of syrup in skillet. Cover and let rise until doubled, about 45 minutes.
Bake in preheated 350°F oven for 23 to 25 minutes or until lightly browned. Cool skillet for 10 minutes on a wire rack, do not remove cinnamon rolls from skillet.
Combine butter, vanilla, and sugar in a small bowl, adding enough milk to reach desired consistency. Drizzle or spread frosting over warm cinnamon rolls while still in skillet. Enjoy!