Make the dough. In a large saucepan warm the milk, vegetable oil, and sugar over medium heat until it reaches 110°F (do not let the mixture boil). Turn off the heat. Sprinkle the yeast over top and let it sit until beginning to foam, about 2 minutes.
Transfer the mixture to the bowl of a stand mixer fitted with a dough hook. Add 4 cups of flour and mix until incorporated. Cover the bowl with a clean kitchen towel and set aside in a warm place, until the dough looks bubbly and has risen, about an hour (Alternatively, you can add the flour directly to the milk mixture in the saucepan and mix the dough with a wooden spoon).
Add the baking powder, baking soda, salt, and remaining ½ cup of flour and stir to combine. Use the dough right away or cover the bowl with plastic wrap and store in the refrigerator for up to 3 days. (Note: If the dough rises over the top of the bowl at any time, punch it down.)
Turn the dough out onto a lightly floured surface. Using a lightly floured rolling pin, roll the dough into a large rectangle about 30 x 10 inches. Spread the butter evenly over the dough, leaving a 1-inch border around the edges. Sprinkle the cinnamon and sugar over the butter.
Starting with the long side closest to you, pull the edge up and over the filling and carefully roll the dough away from you into a log, keeping it fairly tight. Pinch the edge to seal. Using a very sharp knife, cut through the roll, making about 20 rolls, each about 1½-inches wide.
Place the rolls cut-side up into a generously greased 9 x 13 inch baking dish and a 9-inch pie pan, keeping about 1-inch between each roll. Cover with a thin kitchen towel and let rise in a warm place until doubled in size, about 20-30 minutes.
Preheat the oven to 375°F with a rack in the center position.
Bake the rolls for 15 to 20 minutes, or until golden brown and cooked through.
Meanwhile, make the maple icing. In a medium bowl, whisk together the powdered sugar, vanilla soy milk, melted butter alternative, salt, maple flavoring, and coffee, if using, until very smooth. The icing should be pourable.
Drizzle the icing over the warm rolls and spread to evenly coat. Wait a few minutes for the icing to soak in before serving.