In a large bowl, mix together 1 cup of warm milk, 2 + 1/4 tsp. active dry yeast, 1 large egg, 1 Tbsp. melted butter, 4 Tbsp. granulated sugar, and 1/2 tsp. salt. Whisk the mixture quickly. Then, add in 3 cups of all-purpose flour and combine it with a wooden spatula. Add another 1 Tbsp. flour until the dough is not sticky. Knead the dough by hand for about 3 minutes. Cover the bowl with plastic wrap and let it rise in a warm place for about 1 hour.
While the dough is rising, prepare the cinnamon filling. Combine 6 Tbsp. softened butter, 1/4 cup brown sugar, and 1 Tbsp. ground cinnamon in a bowl.
After the dough has risen, lightly dust your work surface with flour and turn the dough out onto the surface. Roll the dough into a 17 x 10 rectangle. Spread the cinnamon filling evenly over the dough, then roll it up lengthwise. Use dental floss to cut the roll into 8 even pieces.
Place the cinnamon rolls into a 9 x 13 pan, cover them with plastic wrap, and let them rise in a warm place for an additional 30 minutes. Once they have risen, brush the tops with the remaining 1 Tbsp. melted butter. Bake at 350 degrees F for 20-25 minutes. Let the rolls cool for about 15 minutes.
While the rolls are cooling, make the frosting. In a medium bowl, beat together 4 Tbsp. softened butter, 4 oz. softened cream cheese, and 2 tsp. vanilla extract using a hand mixer. Gradually add in 1 cup of powdered sugar and mix until smooth.
Once the cinnamon rolls are slightly warm, spread the frosting over the top and enjoy!