Warm the milk to 110°F using a thermometer. In your stand mixer bowl, whisk together the flour, sugar, instant yeast, salt, and cocoa powder until evenly distributed.
Add the warmed milk, melted butter, egg, red food coloring, vanilla extract, and white vinegar to the dry ingredients. Mix on low speed until a shaggy dough forms, about 1 minute.
Increase speed to medium and knead for 6-8 minutes until the dough is smooth, elastic, and slightly tacky but not sticky. If kneading by hand, knead for 10-12 minutes.
Shape the dough into a smooth ball and place it in a large, lightly greased bowl. Turn once to coat all sides, cover with a clean kitchen towel or plastic wrap, and let rise in a warm, draft-free spot for 1-1.5 hours until doubled in size.
While the dough rises, mix together the softened butter, brown sugar, cinnamon, and cocoa powder in a medium bowl until a smooth, spreadable paste forms with no lumps. Set aside at room temperature.
Once risen, punch down the dough to release air bubbles. Turn it out onto a lightly floured surface and roll into a large rectangle approximately 16x12 inches and about ¼ inch thick.
Spread the cinnamon filling evenly over the entire surface of the dough, leaving a ½-inch border along the top long edge. Press the filling gently into the dough to help it adhere.
Starting from the long edge closest to you, tightly roll the dough away from you into a log. Pinch the seam closed and position it seam-side down.
Using a sharp knife or unflavored dental floss, mark the log into 12 equal pieces (about 1.5 inches each). Cut cleanly without squashing the rolls and arrange them in a greased 9x13-inch baking dish.
Cover the baking dish loosely with plastic wrap or a kitchen towel and let the rolls rise for 30-45 minutes until puffy and nearly doubled in size. Near the end of rising time, preheat your oven to 350°F.
Remove the cover and bake for 23-27 minutes until the rolls are set and the tops spring back lightly when touched. The centers should look cooked through, not doughy.
Remove from oven and let cool for 10 minutes. Meanwhile, beat the softened cream cheese and butter together with an electric mixer until smooth and creamy, about 2 minutes.
Add the powdered sugar, vanilla extract, and a pinch of salt. Beat on low speed until combined, then increase to high and beat for 2-3 minutes until light and fluffy. Add milk one tablespoon at a time until desired consistency.
While the rolls are still warm (but not hot), spread or pipe the cream cheese frosting generously over the top. Serve immediately while warm and gooey for the best experience.