In a stand mixer bowl, add yeast, 1 cup flour, sugar and salt; whisk to combine.
Add water, milk and butter to pot or microwave-safe dish; heat to 120º to 130º F.
Add warmed milk mixture to mixing bowl. Using a paddle attachment, beat 4 minutes on medium speed. Add egg; beat 1 minute.
Switch to dough hook attachment. Gradually add remaining flour and knead until dough is smooth and elastic, about 5 to 7 minutes. Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until double in size.
Make filling: Mix together brown sugar, sugar and cinnamon.
Turn dough onto lightly floured surface; punch down to remove air bubbles. Roll or pat into a 15x12-inch rectangle. Spread softened butter evenly over dough. Sprinkle cinnamon-sugar mixture evenly over buttered dough.
Starting with shorter side, roll up tightly. Pinch edges to seal. Cut into 12 slices. Place in greased 13x9-inch cake pan. Cover; let rise until indentation remains after touching.
Preheat oven to 375º F.
Bake for 20 to 25 minutes. Remove from pan; cool.