Mix Dry Dough Ingredients – In the bowl of a stand mixer or large bowl, whisk together the flour, yeast, and salt. Set aside.
Warm Butter & Milk – In a medium microwave-safe bowl, combine the sugar, butter, water, and milk. Microwave on high power in 20-second intervals until it reaches 120 to 130°F (49 to 54°C), about 40 to 60 seconds. Let it cool if it gets too hot. Stir to combine.
Knead & Rest Dough – Add 1 large egg to the flour mixture. Use a spatula to gradually stir in the warm liquid mixture until a soft, sticky dough forms.
Using the dough hook attachment of a stand mixer, mix on low speed (setting 2) until the dough becomes smooth and elastic, about 4 minutes. If not using a mixer, knead by hand on a lightly floured surface for about 5 minutes. Cover the dough with plastic wrap and rest on the counter for 10 minutes. Meanwhile, make the filling.
Make the Filling – In a medium bowl, stir together the brown sugar, granulated sugar, flour, and cinnamon. Add 6 tablespoons of softened butter and mix until smooth and spreadable.
Roll Out Dough – On a lightly floured surface, roll out the dough into a 16 x 9-inch rectangle.
Spread on Filling – Evenly spread the filling over the dough, leaving a 1⁄4-inch border on all sides. Tightly roll the dough into a 16-inch long log. Trim off a small piece of dough from each end. Cut the log into 12 even-sized pieces.
Let Rolls Rise – Place the rolls in a lightly greased 9 x 13-inch baking dish, spaced out about 1⁄2-inch apart, 3 rolls per row. Leave about 1⁄2-inch of space along the edges of the pan. Cover with plastic wrap and allow to rise in a warm draft-free area or proofing box (see notes) until doubled in size, about 30 minutes.
Bake – Adjust the oven rack to the lower-middle position. Heat to 375°F (191°C). Uncover and bake the rolls until lightly golden brown in color, about 15 to 17 minutes. Slightly cool the rolls for 10 minutes. Meanwhile, make the glaze.
Make the Glaze – In a medium bowl, whisk the powdered sugar, warm milk, vanilla extract, and lemon juice until smooth. Drizzle glaze over the warm cinnamon rolls. Serve immediately.